The elliptic stochastic quantization of some two dimensional Euclidean QFTs
Sergio Albeverio Francesco C. De Vecchi Massimiliano Gubinelli








`=<
T160


















theoremTheorem
corollary[theorem]Corollary
definition[theorem]Definition
lemma[theorem]Lemma
proposition[theorem]Proposition
remark[theorem]Remark








enumeratenumeric 
itemizedot 














a4paper,

left=30mm,
right=30mm,
top=30mm,













  The elliptic stochastic quantization

  of some two dimensional Euclidean QFTs

    
  Sergio Albeverio, Francesco C. De Vecchi and Massimiliano Gubinelli
  Hausdorff Center of Mathematics &

  Institute of Applied Mathematics, 

  University of Bonn, Germany
  albeverio@iam.uni-bonn.de
  francesco.devecchi@uni-bonn.de
  gubinelli@iam.uni-bonn.de

    December 30, 2023
=============================================================================================================================================================================================================================================================================




  We study a class of elliptic SPDEs with additive Gaussian noise on
  ℝ^2 × M, with M a d-dimensional manifold equipped with a
  positive Radon measure, and a real-valued non linearity given by the
  derivative of a smooth potential V, convex at infinity and growing at most
  exponentially. For quite general coefficients and a suitable regularity of
  the noise we obtain, via the dimensional reduction principle discussed in
  our previous paper <cit.>, the identity between the
  law of the solution to the SPDE evaluated at the origin with a Gibbs type
  measure on an abstract Wiener space over M. The results are then applied
  to the elliptic stochastic quantization equation for the scalar field with
  polynomial interaction over 𝕋^2, and with exponential interaction
  over ℝ^2 (known also as Høegh-Krohn or Liouville model in the
  literature). In particular for the exponential interaction case, the
  existence and uniqueness properties of solutions to the elliptic equation
  over ℝ^2 + 2 is derived as well as the dimensional reduction
  for the values of the “charge parameter” σ = α/2
  √(π) < √(4 ( 8 - 4 √(3)) π)≃√(4.23
  π), for which the model has an Euclidean invariant probability measure
  (hence also permitting to get the corresponding relativistic invariant model
  on the two dimensional Minkowski space).


60H1581T0881T40

stochastic quantizationelliptic stochastic partial
differential equationsdimensional reductionEuclidean
quantum field theoryexponential interaction





§ INTRODUCTION


The Euclidean approach to quantization of relativistic non-linear wave
equations requires the construction of certain probability measures supported
on distributions satisfying a set of quite strong requirements among which
invariance under the full group of rigid motions of the Euclidean space and
reflection positivity. Such constructions have succeeded in the case where the
dimension d of the Euclidean space is equal to 1, 2, 3 (that is also the
dimension of the space-time of the original relativistic equation). The
non-linearity is quite general if d = 1, whereas it is restricted to the
derivatives of polynomials of even degree respectively suitable superpositions
of trigonometric or exponential functions for d = 2, or a cubic monomial
function for d = 3 (the φ^4_3 model). In these cases a number of
interesting physical and mathematical properties of the quantized relativistic
fields models have been put in evidence.

In recent years new methods coming from the study of singular stochastic
partial differential equations (SSPDEs) have been developed and have opened
the possibility of new constructions of such Euclidean measures (and other
similar measures associated with problems coming from other areas of
applications, like statistical mechanics, hydrodynamics, wave propagation in
random media ). In these approaches the relevant Euclidean
invariant (probability) measures are obtained as invariant measures for
certain parabolic semilinear SPDEs called stochastic quantization equations
(SQEs).

Typically the measures to be constructed have the heuristic form

    μ (φ) = Z^- 1 e^- S (φ)𝒟φ,

where φ = φ (x) ∈ℝ, x ∈ℝ^d,
𝒟φ = Π_x ∈ℝ^dφ (x) is a
“flat measure”, Z is a “normalization constant”, S (φ) = S_0
(φ) + λ S_int (φ) with S_0 (φ) =
1/2( ∫_ℝ^d | ∇φ (x) |^2  x + m^2
∫_ℝ^d | φ (x) |^2  x ), and where λ⩾ 0, m ⩾ 0 are parameters,

    S_int (φ) = ∫_ℝ^d V (φ (x))  x,

for some (non-linear) measurable real valued function V over ℝ.
Note that μ (φ) is heuristically invariant under Euclidean
transformations (translations and rotations and also additional reflections,
in the case where V is an even function).

The associated stochastic quantization equation (first introduced
in <cit.>) is of the form

    X_τ = (Δ - m^2) X_τ t - V' (X_τ) 
       t +  W_τ

where τ is an additional “computer time”, Δ is the Laplacian in
ℝ^d, X_τ = X_τ (x), x ∈ℝ^d, is a random
variable taking values, for fixed τ, in the space of “(generalized)
functions” of x, W_τ (x) is a Gaussian white noise in τ
and x, and V' in the derivative of V.

Following a terminology used especially when V is a polynomial P, we call
the above SPDE the V (φ)_d–stochastic quantization equation (SQE).
Various detailed results have been obtained on P (φ)_2 SQEs, see the
introductions of <cit.>
and <cit.> for references. In particular
in <cit.> there were obtained a strong solution and a
unique ergodic measure in the case where ℝ^2 is replaced by the
2-torus 𝕋^2.

As for the case of the φ^4_3 SQE, a breakthrough for local in time
solutions of the SQE on the 3-torus 𝕋^3 came
from <cit.>, see also <cit.>. More recently still
for the case of the 3-torus 𝕋^3 a construction of invariant
solutions has been given in <cit.>. This can be
looked upon as a construction of the φ^4_3-measure on 𝕋^3,
by methods different from the previous ones provided by mathematical physics
approaches, see references in <cit.>. The methods
in <cit.> have been simplified and extended to the
case of ℝ^3 in <cit.>, yielding a full
alternative construction of the measures which qualify to be called
φ^4_3 Euclidean measures. A previous approach following
<cit.> is in <cit.>. A number of open problems has been
mentioned in <cit.>, the main
one concerns the uniqueness of the invariant measures.

Other approaches are however possible if one understands stochastic
quantization in the broader sense of associating to a given heuristic
“Euclidean measure” a stochastic process having this very measure as
suitable marginal. A first approach consists in constructing a jump type
process via infinite dimensional quasi-regular jump-type Dirichlet
forms <cit.> (for such
Dirichlet forms see also <cit.>), that has by construction the
given Euclidean measure as invariant measure. Another approach obtains the
invariant measure of an SQE in dimension d by looking at solutions of a
suitable associated elliptic SPDE in dimension d + 2 and restricting the
solution to the d dimensional Euclidean space obtained sending to zero the
two additional variables. This latter procedure is known in the physical
literature as Parisi-Sourlas dimensional reduction and has been implemented by
using algebraic methods of supersymmetry (see <cit.>).
For the case of a regularized non-linear term in the original SQE and
regularized noise it has been studied from a mathematical point of view
in <cit.>.

Elliptic stochastic PDEs related to stochastic quantization of the
φ_2^4 and φ_3^4 models respectively have been discussed both
on the torus and in all space in <cit.>. A systematic mathematical
investigation of the mechanism of dimensional reduction has been initiated
in <cit.>, for the V (φ)_0 model, for both the
cases where V is a convex function and also the case where V is
non-convex, where an interesting phenomenon of non-uniqueness of solutions has
been put in evidence.

In fact in <cit.> an explicit formula for the law of
the solution of a class of elliptic SPDE in ℝ^2 taken at the origin
has been obtained, by means of a rigorous version of dimensional reduction.
Besides proving an instance of elliptic stochastic quantization for scalar
fields and for the case of underlying Euclidean dimension d = 0, it also
provided a realization of the relation between a supersymmetric quantum field
model and an elliptic SPDE in 2 dimensions.


The present paper has two principal aims. The first one is to prove the
dimensional reduction principle for elliptic SPDEs with non singular noise for
equations in d + 2 dimensions with d > 0. The second is to extend the
result to elliptic singular SPDE (i.e. with space-time white noise) having
applications in stochastic quantization program and analyzing the
corresponding dimensional reduction with respect to removal of spatial
cut-offs.

We restrict our attention to singular SPDEs with d = 2 and analyze
polynomial and exponential interactions. In particular, in the case of
potentials of the form exp (σφ)_2, for suitable real σ
we are able to complete the dimensional reduction picture removing all the
spatial cutoffs (i.e. in both the “fictitious” and “real” spatial
variables).


Before passing to a more detailed description of our results, let us stress
the importance of the exp (σφ)_2 model (and related ones) in
relativistic and Euclidean quantum field theory. The exp (σφ)_2
model over ℝ^2 has been first introduced by
Høegh-Krohn <cit.> (see also <cit.>
and <cit.>, p. 178 and 307-313) and constructed for an interaction
of the form

    U_g (φ) : = ∫ : e^σφ (z) : g (z)  z ν
       (σ),

   where ν positive bounded measure, for any positive space cut-off function g ∈ L^1 (ℝ^2) ∩ L^2
(ℝ^2) and any positive finite measure ν with support in (-
√(4 π), √(4 π)), in the sense that the interacting measure

    μ_g (φ) : = e^- U_g (φ)μ_0 (φ)/∫ e^- U_g (φ)μ_0 (φ),

for m > 0, is absolutely continuous with respect to Nelson's free field
Gaussian measure μ_0 with mean zero and covariance (m^2 - Δ_z)^-
1, z ∈ℝ^2 (realized as a probability measure on, for example,
𝒮' (ℝ^2)). See <cit.>
and <cit.> p. 178 for further properties of the measure μ_g.
This model has been discussed in the infinite volume limit g → 1,
that is unique, in <cit.>, under the more restrictive assumption
that supp (ν) ⊂[ - 4/√(π),
4/√(π)] (see also <cit.> pp. 307-313 and
<cit.>).


In fact in <cit.> the authors prove that the measure of the exp
(σφ)_2 model for | σ | < √(4 π) (or more generally
of suitable superpositions ∫Ch (σφ)_2 ν
(σ), where supp (ν) ⊂[ - 4/√(π),
4/√(π)], of these models) satisfies all axioms of
Euclidean quantum field theory and leads to relativistic quantum fields over
two-dimensional Minkowski space-time satisfying all Wightman axioms with
interaction and a positive, finite mass gap at the lower end of the spectrum
of the corresponding Hamiltonian.

Further properties of such models were discussed in,
e.g., <cit.>
and <cit.>. The case | σ | <
√(8 π) was discussed by S. Kusuoka in <cit.> and the relation with
independent work on multiplicative chaos by J. P. Kahane <cit.> (see also
below) was pointed out by H. Sato. In <cit.> and
also <cit.> estimates in L^p (for the interaction in a bounded
region) were given for | σ | < √(4 π/p - 1). The Gaussian
character of the model for | σ | sufficiently large or for any σ
if the Euclidean space dimension d satisfies d ⩾ 3  has been
pointed out in <cit.> and <cit.>. The relevance of the exp
(σφ)_2 model for Polyakov string theory has been discussed
in <cit.> (| σ | < √(4 π)
corresponding to the embedding dimension D < 13) and rediscovered more
recently in connection with topics like Liouville model, quantum gravity and
multiplicative chaos. Let us mention in this
connection <cit.> see also
<cit.> (for the literature for
other approaches to Liouville type models not directly connected with
probabilistic methods see, e.g., <cit.>).

Finally a connection of the exp (σφ)_2 model with the
irreducibility of a unitary representation of the group of mappings of a
manifold into a compact Lie group has been pointed out and studied
in <cit.>, see also the recent work <cit.>.


The exp (σφ)_2 model is also closely connected with the sin
(σφ)_2 model (“Sine-Gordon equation”) as discussed
in <cit.>. In the latter reference all Wightman axioms are
proved again in the region | σ | < √(4 π).

Both in the exp (σφ)_2 model and the sin (σφ)_2
model a replacement of the exp (respectively sin) function by a Wick
ordered version is needed (see <cit.>) but also suffices for the
existence and non-triviality of the model for | σ | up to √(4
π). For larger values of | σ |, in the case of the sin (σφ)_2 model, up to √(8 π), further renormalization by
counter-terms is required (see <cit.>).


The study of the stochastic quantization equation associated with the latter
class of models has been initiated in <cit.> (where strong solutions
have been discussed and the necessity of renormalization has been pointed
out). In the case where ℝ^2 (or 𝕋^2) is replaced by
ℝ (respectively 𝕋), i.e. for the model exp (σφ)_1, a deeper analysis is possible and has been pursued
in <cit.>, where the existence of solutions and the strong uniqueness
of the invariant measure are proven (for the corresponding stochastic
quantization of P (φ)_1 see <cit.>).

The case of the SQE in d = 2 and with a regularized noise on the torus (and
corresponding changes in the coefficients of the stochastic quantization
equation needed to keep the same invariant measure) has been discussed
in <cit.> (see also <cit.> for further developments), where
existence of solutions was proven using essentially the properties of the
exp (σφ)_2 model established in <cit.> and
methods of <cit.>. Uniqueness problems are also
discussed in <cit.> in conjunction with an approach using Dirichlet
forms, for all σ^2 < 4 π, in a setting similar to <cit.> (see
the introduction of <cit.> for further references).

Hairer and Shen <cit.> introduced powerful methods to handle the
dynamical sin (σφ)_2 model on 𝕋^2, and via
regularity structures local existence is shown up to σ^2 < 16/3π. More recently these results has been extended in <cit.> to
cover all the subcritical regime σ^2 < 8 π.


Concerning the exponential interaction, the work of
Garban <cit.> appeared recently in which the author studies the
SQE on the torus 𝕋^2 and on the sphere 𝕊^2. After
subtracting the solution to the linear equation Garban obtains an SPDE driven
by a multi-fractal and intermitted multiplicative chaos. When | σ | <
( 4 - 2 √(3)) √(π)( < √(4 π)) (a
regime correspondingly called Da Prato-Debussche phase) he shows the existence
of a strong solution and the convergence of the solution to the equation with
regularized noise to the singular one both on the torus 𝕋^2 and on
the sphere 𝕊^2 (see Theorem 1.7 and Theorem 1.9
in <cit.>). The method used is based on the Besov regularity of
the Gaussian multiplicative chaos (related to the theory of <cit.>). The
paper <cit.> also refers to other interesting relations to
quantum gravity, conformal fields theory, strings theory, multiplicative chaos
and exciting new results on random measures. A result on existence and
uniqueness for solutions of SQE (without the proof of the convergence of the
solution to the equation with regularized noise to the singular one) is also
proved (Theorem 1.11 in <cit.>) for | σ | < ( 4 - 2
√(2)) √(π)( < √(4 π)). Moreover a
comparison with the SQE for the Sine-Gordon model is provided (cfr. Section 7
in <cit.>).[After the submission of this article for
publication we have become aware of the papers
<cit.> which address the stochastic
quantization of the exponential model.]


Let us briefly review and comment the main results obtained in the present
paper. In Section <ref> we study the case of the following
elliptic SPDE (equation (<ref>) in Section <ref>)

    (- Δ_x + m^2 + 𝔏) (ϕ) (x, z) + 𝒜^2 [f g
       ∂ V (ϕ)] (x, z) = ξ^A (x, z)

on ℝ^2 × M, with M a d-dimensional manifold with or
without boundary, equipped with a positive Radon measure z and a
real-valued non-linearity given by the derivative ∂ V of a positive smooth
function V defined on the real line growing at most exponentially at
infinity. We use the shorthand f g V'(ϕ) to mean the function f g ∂ V(ϕ): ℝ^2× M  ⟶ℝ such that (x, z) ⟼ f (x) g (z) ∂ V
(ϕ (x, z)) (where x∈ℝ^2 and z∈ M).











 Additional assumptions are as follows. The function V is
taken to the convex or “quasi convex” (see Hypothesis QC below).
𝔏 and 𝒜 are two positive self-adjoint operators
acting on L^2 (M), 𝒜 is bounded, 𝔏 is
not-necessarily bounded but has a dense domain in L^2 (M), 𝒜 and
𝔏 commute and both have completely discrete spectrum with some
additional hypotheses on the eigenfunctions and the eigenvalues of
𝒜. The noise ξ^A is supposed to be Gaussian with mean zero and
covariance corresponding to a regularization given by 𝒜. Finally
f is a C^2 real-valued cut-off function on ℝ^2, decaying
exponentially at infinity and g is a smooth real-valued cut-off with compact
support on M.

We introduce finite dimensional projections relative of (<ref>)
and prove the existence of a weak solution ν satisfying a form of the
dimensional reduction principle described in our previous
paper <cit.>, extended to the d dimensional case
with the presence of a non-trivial operator 𝔏. See
Theorem <ref> for a precise statement.

Next we prove extensions of these results in two directions: in
Section <ref> we consider the case M =ℝ^d,
𝔏 = - Δ_z (cfr. Theorem <ref>), in
Section <ref> we consider the problem of the removal of
the cutoffs f, g in the case where V is a convex function (cfr.
Theorem <ref>).


The first part of the present paper contains the first example of rigorous
dimensional reduction for elliptic SPDE in d + 2 dimension with d > 0.
Indeed, the only paper which to our knowledge study dimensional reduction from
the rigorous point of view before our work was <cit.> whose main
result is a theorem about the dimensional reduction of an integral in a space
of functions in d + 2 variables to a Gibbs type measure in a space of
functions in d variables. In particular the authors do not attack the
problem of the relation with the elliptic SPDE. Furthermore
in <cit.> only polynomial potentials V are considered and the
regularization of the noise is chosen to be compactly supported in Fourier
space. Our Hypothesis QC on the potential V and Hypotheses H𝒜
and H𝒜 1 for the regularization of the noise considered here are
quite more general. Let us stress that the results obtained under these more
general hypotheses use in essential way the SPDE formulation of dimensional
reduction.


In Section <ref> we extend our results to singular SPDEs.
In particular the elliptic stochastic quantization of the exponential
interaction is proven, where V is a suitable renormalized exponential
function exp (αϕ - ∞), for | α | < α_max, where 

    α_max= 4 √(8 - 4
    √(3))π,

(corresponding to the reduced index | σ | <α_max/2√(π)= √(4
( 8 - 4 √(3)) π)), and the white noise is unregularized.
Existence and uniqueness is first proven for a suitable regularized model
using the results of the previous sections (cfr.
Theorem <ref>). Subsequently the removal of the
regularization in the noise is achieved in the Besov space B^s_p, p, ℓ
(ℝ^4), where - 1 < s < 0 and 1 < p ⩽ 2 are suitable
constants depending only on α. Using the existence, uniqueness and
convergence results for the elliptic SPDEs over ℝ^4 proven in
Section <ref>, the existence and uniqueness results for
the elliptic stochastic quantization equation for exponential interaction
follows in Theorem <ref>. The dimensional reduction
result is then obtained (cfr. Theorem <ref>)
with uniqueness for all | α | < α_max. Our last
result in this section, Theorem <ref>, shows the
convergence and uniqueness of the equation when the spatial cut-off g is
removed. As an easy corollary one obtains the full Euclidean invariance of the
law of the solutions.

Finally in Section <ref> we prove the existence for singular
elliptic SPDE with Wick power non linearity in d + 2 = 4 dimension.
Furthermore we prove a dimensional reduction principle for some weak solutions
to SPDEs of this form providing the first example of elliptic stochastic
quantization for the polynomial interaction model P (φ)_2. Let us
stress that for polynomial interactions the problem of uniqueness of weak
solutions to the elliptic SPDE is open and this prevents a more detailed
analysis of the dimensional reduction.


The results contained in this second part of the paper gives the first
example of dimensional reduction for singular elliptic SPDE as originally
formulated by Parisi and Sourlas in <cit.>. In particular
with respect to <cit.>, where the existence for singular elliptic SPDEs
with polynomial type non-linearity is proven for both d + 2 = 4 and, with
cubic type non-linearity, d + 2 = 5 dimension, in
Section <ref> we prove, at least in the d + 2 = 4 dimensional
case, that they can be used as stochastic quantization equation for quantum
field theory. However many problems still remain open, for the polynomial
models, with respect to the removal of the spatial cutoffs.


Let us stress that, in Section <ref>, we prove existence
and uniqueness of solutions for elliptic SPDE and the relation with the
Liouville measure in the regime | σ | < α_max/2√(π). This result is achieved using in an essential way two
properties of the exponential model: the fact that the Wick exponential of a
distribution is a positive measure (this fact is already exploited
in <cit.>) and the multifractality of Wick exponentials (see
Lemma <ref> for a precise formulation of this property). In
particular we prove the existence and uniqueness for the elliptic SPDE with
Wick exponential non-linearity using the space B^s_p, p, ℓ
(ℝ^4), instead of B^s'_∞, ∞, ℓ (ℝ^4),
and by only the Da Prato-Debussche trick. This is possible since for any
exponent | σ | <α_max/2√(π)= √(4 ( 8 - 4 √(3)) π) the noise
lives, beside that, inside B^s_p, p, ℓ (ℝ^4) for some - 1 < s
< 0 and 1 < p ⩽ 2, instead for | σ | ⩾( 4 - 2
√(2)) √(π)( < √(4 π) < α_max/2√(π)) the Wick power of the noise is in B^s_∞,
∞, ℓ (ℝ^4) with s < - 2. The other novelty is the fact
that we are able to solve the equation in the full space and we are then
easily able to prove the Euclidean invariance of the law.

The other best result, to our knowledge and before our paper, concerning
stochastic quantization of exponential interaction, is the
paper <cit.>, which studied the stochastic quantization in the
parabolic setting for the charge parameter | σ | < ( 4 - 2 √(2)) √(π)( < √(4 π)). We think that the analytic
methods, developed here in Section <ref>, joined with the
probabilistic results, typical of the parabolic setting obtained
in <cit.>, can also be applied to the parabolic case for the
regime | σ | < α_max/2√(π).

After the submission of this article for publication we became aware of
<cit.> that address the stochastic quantization of the
exponential model in the parabolic setting on the two dimensional torus
𝕋^2 for σ^2 < 4 π and σ^2 < 8 π respectively. Let
us mention also the papers, which have appeared after the submission of the
present article, <cit.> where the stochastic quantization of the
exponential model is addressed in the parabolic and hyperbolic setting on the
two dimensional torus 𝕋^2 and for σ^2 < 4 π, and
<cit.> where the parabolic stochastic quantization of the model
related to Liouville quantum gravity on a general compact Riemannian manifold
is treated in the L^2 regime. Our paper seems to be the only one studying
the elliptic case, or more generally the exponential model on the whole
ℝ^2.  

Acknowledgments. The authors are grateful to an anonymous referee
of the paper, for the careful reading of the article and for the constructive
comments that have improved very much the presentation of the results. The
first named author is grateful to HCM of the University of Bonn, to Stefania
Ugolini (Università degli Studi di Milano) and to the Department of
Mathematics of Università degli Studi di Milano for financial support.
This research has been supported by the German Research Foundation (DFG) via
CRC 1060.



§ DIMENSIONAL REDUCTION WITH REGULARIZED NOISE




 §.§ Discrete spectrum


Consider the following elliptic SPDE

    (- Δ_x + m^2 + 𝔏) (ϕ) (x, z) + 𝒜^2 [f g
     ∂ V (ϕ)] (x, z) = ξ^𝒜 (x, z)


where x ∈ℝ^2 and z ∈ M (where M is a d dimensional
manifold with or without boundary, equipped with a Radon positive measure
z) and V is a smooth function on ℝ growing at most
exponentially at infinity, ∂ V its gradient, ϕ : ℝ^2
× M →ℝ a scalar random field, and the function f g
 ∂ V (ϕ) is defined as follows

    [                f g ∂ V(ϕ):                     ℝ^2× M                          ⟶                          ℝ;                                                (x, z)                          ⟼ f (x) g (z) ∂ V
    (ϕ (x, z)) ]



We now list the hypotheses and assumptions on the various elements of
equation (<ref>) and of its generalization to the vector case,
where V is replaced by V_n being defined on ℝ^n (when
ℝ^n =ℝ hereafter instead of writing V_1 we simply write
V).


  
  Hypothesis C. The potential V_n : ℝ^n →ℝ is a positive smooth function such that
  
    y ∈ℝ^n ↦ V_n (y),

  is strictly convex and it and its first and second partial derivatives grow
  at most exponentially at infinity.
  
  
  Hypothesis QC. The potential V_n : ℝ^n →ℝ is a positive smooth function, such that it and its first and
  second partial derivatives grow at most exponentially at infinity and
  additionally such that there exists a function ℌ : ℝ→ℝ with exponential growth at infinity such that we
  have
  
    - ⟨n̂, ∂ V_n (y + r n̂) ⟩⩽ℌ (y),     n̂∈𝕊^n - 1, y ∈ℝ^n and r ∈ℝ_+,

  with 𝕊^n - 1 is the n - 1 dimensional sphere.
  
  
  Hypothesis Hf. The non-negative function f : ℝ^2
  →ℝ, is invariant with respect to rotations (i.e. there
  exists f̃ : ℝ_+ →ℝ_+ such that f (x)
  = f̃ (| x |^2)), it has at least C^2 smoothness and in addition
  satisfies f' (x) = f̃' (| x |^2) ⩽ 0, it decays
  exponentially at infinity and fulfills Δ (f) ⩽ b^2 f for b^2
  ≪ m^2 (some examples of such functions are given in <cit.>).
  
  
  Hypothesis Hg. The non-negative function g : M →ℝ is smooth and with compact support.
  
  
  Hypothesis H𝔏. The operator 𝔏 is closed
  (possibly unbounded) and defined on the vector space D_𝔏⊂ L^2 (M) (where L^2 (M) is the space of measurable L^2 functions
  defined on M with respect to the measure z on M), which is
  dense in L^2 (M). The range of 𝔏 is a subset of L^2 (M).
  The operator 𝔏 is positive and self-adjoint and has a
  completely discrete spectrum. Furthermore we suppose that there exists at
  least an orthonormal basis H_1, …, H_k, … in L^2 (M) composed
  by eigenfunctions of 𝔏 which are C^0_𝔴 (M)
  functions (where C^0_𝔴 (M) is the space of continuous
  functions h defined on M such that h (z) 𝔴 (z)
  _∞ < + ∞ where 𝔴 is a positive continuous
  function uniformly bounded from above). Finally we denote by λ_1
  ⩽λ_2 ⩽⋯⩽λ_k ⩽⋯
  the eigenvalues of 𝔏 corresponding to the eigenfunctions H_1,
  H_2, …, H_k, ….
  
  
  Hypothesis H𝒜. The operator 𝒜 : L^2 (M)
  → L^2 (M) is a linear, injective, continuous, positive
  self-adjoint operator commuting with 𝔏. If we denote by
  σ_1, …, σ_k, … the eigenvalues of 𝒜
  corresponding to the basis H_1, …, H_k, … we suppose that
  
    ∑_k = 1^∞σ_k < + ∞∑_k =
         1^∞σ_k  H_k _∞, 𝔴^2 < + ∞,

  where ·_∞, 𝔴 denotes the L^∞-norm in
  the weighted space C^0_𝔴 (M).
  
  
  Hypothesis Hξ. The noise ξ^𝒜 is Gaussian with
  mean zero and a covariance corresponding to a regularization provided by
  𝒜, namely such that if h_1, h_2 ∈ C^∞_0 (ℝ^2
  × M) we have
  
    𝔼 [⟨ h_1, ξ^𝒜⟩⟨ h_2,
         ξ^𝒜⟩] = ∫_ℝ^2 × M (I ⊗𝒜) (h_1) (x, z) (I ⊗𝒜) (h_2) (x, z)  x
          z.

     
     
Hereafter when V is a scalar function (i.e. V:ℝ→ℝ) we use the notation V':=∂ V.
     

  When V_1 = V : ℝ→ℝ, Hypothesis QC can be
  reformulated as
  
    ∓ V' (y ± r) ⩽ℌ (y),

  where r ⩾ 0.


In the rest of this Subsection <ref> we will always assume
Hypothesis Hf, Hg, H𝔏, H𝒜 and we specify the use
of Hypothesis QC or C case by case. Moreover, hereafter we systematically use
the following abuse of notation: we denote by 𝔏 and 𝒜
both the operators with the same name defined on the space L^2 (M) and also
the operators I ⊗𝔏 and I ⊗𝒜 defined on
L^2 (ℝ^2 × M).


We describe now the setting for equation (<ref>). We assume that
ξ^𝒜 is defined on an abstract Wiener space (𝒲,
ℋ, μ^𝒜) where the Cameron-Martin space is

    ℋ = L^2 (ℝ^2) ⊗_H 𝒜 (L^2 (M)),

(here ⊗_H is the natural tensor product of Hilbert spaces) equipped
with the following scalar product

    ⟨ h_1, h_2 ⟩ = ∫_ℝ^2 × M𝒜^- 1
       (h_1) (x, z) 𝒜^- 1 (h_2) (x, z)  x  z,      h_1,
       h_2 ∈ℋ .

The Wiener space 𝒲 is given by

    𝒲    =     (- Δ_x + 1) (C^0_ℓ (ℝ^2) ∩ W^1 -,
      p_ℓ (ℝ^2)) ⊗_ϵ𝒜^1 / 2 (L^2 (M)),

where ℓ⩾ 0,  C^0_ℓ (ℝ^2) is the space of
continuous functions on ℝ^2 such that

    k _∞, ℓ = sup_x ∈ℝ^2 (| k (x) | (1 + | x
       |^2)^- ℓ / 2),

W^1 -, p_ℓ (ℝ^2) denotes the Sobolev space of regularity 1
- and weight

    r_ℓ (x) = (1 + | x |^2)^- ℓ / 2,

and ⊗_ϵ denotes the injective tensor product whose norm is
given by

    w _𝒲    =    sup{| ∑_i = 1^n ⟨𝔟, k_i ⟩_𝒪_ℓ⟨𝔠, h_i
      ⟩_𝒜^1 / 2 (L^2 (M))|, 𝔟∈ℬ_𝒪_ℓ^  𝔠∈ℬ_𝒜^1 / 2 (L^2 (M))^}
        =    sup{∑_i = 1^n ⟨𝔟, k_i
      ⟩_𝒪_ℓ h_i _𝒜^1 / 2 (L^2 (M)),
      𝔟∈ℬ_𝒪_ℓ^}
        =    sup{∑_i = 1^n ⟨𝔠, h_i
      ⟩_𝒜^1 / 2 (L^2 (M)) k_i _𝒪_ℓ,
      𝔠∈ℬ_𝒜^1 / 2 (L^2 (M))^}

where we suppose that w = ∑_i = 1^n k_i ⊗ h_i ∈𝒲,
𝒪_ℓ = (- Δ_x + 1) (C^0_ℓ (ℝ^2)
∩ W^1 -, p_ℓ (ℝ^2)), ⟨·, ·⟩_𝒪_ℓ and ⟨·,·⟩_𝒜^1 / 2 (L^2 (M)) are the natural duality of
the Banach spaces 𝒪_ℓ and 𝒜^1 / 2 (L^2 (M))
respectively and ℬ_𝒪_ℓ^ and
ℬ_𝒜^1 / 2 (L^2 (M))^ are the unit balls
of 𝒪_ℓ^ and (𝒜^1 / 2 (L^2
(M)))^ respectively (i.e., the natural dual Banach spaces of
𝒪_ℓ and 𝒜^1 / 2 (L^2 (M));
see <cit.> for more details). In particular if w = ∑_i =
1^∞ k_i ⊗ h_i (where the series is supposed to converges in
𝒲 with respect to its natural strong topology) we have

    w _𝒲⩽sup_τ∈ℬ_𝒜^- 1
      / 2 (L^2 (M))√(( ∑_i = 1^∞( ∫_M τ (z)
      h_i (z)  z )^2 ))√(∑^∞_i = 1 k_i
      ^2_𝒪_ℓ),

where ℬ_𝒜^- 1 / 2 (L^2 (M)) is the unit ball of
𝒜^- 1 / 2 (L^2 (M)) (here we exploit the fact that
𝒜^- 1 / 2 (L^2 (M)) is the dual space of 𝒜^1 / 2
(L^2 (M)) with respect the standard scalar product of L^2 (M)), we use
Cauchy-Schwarz inequality and the fact that 𝒜^- 1 / 2 (L^2 (M))
is the dual of 𝒜^1 / 2 (L^2 (M)) with respect to the scalar
product of L^2 (M).


The measure μ^𝒜 is the centered Gaussian measure with
Cameron-Martin space ℋ.

In order to prove that (𝒲, ℋ, μ^𝒜) is
actually an abstract Wiener space it is sufficient to prove that 𝔼
[ξ^𝒜_𝒲] < + ∞ where ·_𝒲 denotes the norm of 𝒲. We have that

    ξ^𝒜 (x, z) = ∑_k = 1^∞σ_k ξ^k (x) H_k (z),

where x ∈ℝ^2, z ∈ M, ξ^k are a sequence of independent
Gaussian white noises defined on ℝ^2 and σ_k are defined as
in Hypothesis H𝒜.

Using inequality (<ref>) we have that

    ξ^𝒜_𝒲   ⩽   sup_τ∈ℬ_𝒜^- 1 / 2 (L^2 (M))√(( ∑_k =
      1^∞σ_k ( ∫_M τ (z) H_k (z)  z )^2
      ))
          ·√(∑^∞_k = 1σ_k  (- Δ_x + 1)^- 1 (ξ^k)
      ^2_C^0_ℓ (ℝ^2) ∩ W^1 -, p_ℓ (ℝ^2))
       ≤   √(∑^∞_k = 1σ_k  (- Δ_x + 1)^- 1
      (ξ^k) ^2_C^0_ℓ (ℝ^2) ∩ W^1 -, p_ℓ
      (ℝ^2)),

where in the second inequality we use the fact that √(( ∑_k =
1^∞σ_k ( ∫_M τ (z) H_k (z)  z )^2
)) = 1 whenever  τ∈ℬ_𝒜^- 1 / 2 (L^2
(M)).


On the other hand, by Hypothesis H𝒜, we obtain

    𝔼[ ∑^∞_k = 1σ_k  (- Δ_x + 1)^- 1
       (ξ^k) ^2_C^0_ℓ (ℝ^2) ∩ W^1 -, p_ℓ
       (ℝ^2)] ≲∑_k = 1^∞σ_k < + ∞,

(where ≲ stands for ⩽ modulo a multiplicative positive
constant), and thus 𝔼 [ξ^𝒜_𝒲] < +
∞. This proves that (𝒲, ℋ, μ^𝒜) is
an abstract Wiener space when ξ^𝒜 : 𝒲→𝒲, defined as ξ^𝒜 (w) = w, has exactly
μ^𝒜 as probability distribution.

For later use, it is important to note that 𝒲 is continuously
embedded in

    𝒲' = (- Δ_x + 1) (C^0_ℓ (ℝ^2) ∩ W^1 -,
       p_ℓ (ℝ^2)) ⊗_ϵ C^0_𝔴 (M),

since 𝒜^1 / 2 (L^2 (M)) is continuously embedded in
C^0_𝔴 (M) by Hypothesis H𝒜. Furthermore by
definition of injective tensor product we have 𝒲' ⊂
C^0_r_ℓ (x) 𝔴 (z) (ℝ^2 × M), with r_ℓ
(x) is defined in (<ref>), indeed

    w _C^0_r_ℓ (x) 𝔴 (z)    =    sup_x_0 ∈ℝ^2, z_0 ∈ M | ⟨ r_ℓ (x_0) 𝔴 (z_0)
      δ_x_0⊗δ_z_0, w ⟩ | 
       ⩽   sup{ | ⟨𝔟⊗𝔠, w
      ⟩ |, 𝔟∈ℬ_𝒪_ℓ^  𝔠∈ℬ_C^0_𝔴 (M)^} =  w _𝒲',

where δ_x_0 and δ_z_0 are Dirac deltas in x_0 ∈ℝ^2 and z_0 ∈ M respectively (see <cit.> Section 3.2
for the details of the proof).



Equation (<ref>) can be written as a problem defined on the
abstract Wiener space (𝒲, ℋ, μ^𝒜). In
particular we define the map U : 𝒲→ℋ by

    U (w) = 𝒜^2 [f (x) g (z) V' (ℐ w)],

where w ∈𝒲, x ∈ℝ^2, z ∈ M, f and g as in
equation (<ref>), and ℐ : 𝒲→ (-
Δ_x + 1)^- 1 (𝒲) is the linear operator given by
ℐ = (- Δ_x + m^2 + 𝔏)^- 1. We define the map T
: 𝒲→𝒲 as T (w) = w + U (w).


  It is clear that if S is any measurable map S :
  𝒲→𝒲 satisfying T ∘ S =
  id_𝒲 μ^𝒜-almost surely we have that
  ϕ (w, x, z) = ℐ (S (w)) (x, z) is a (strong) solution to
  equation (<ref>). Furthermore if ν is a probability law on
  𝒲 such that T_ (ν) = μ^𝒜 then a
  random variable ϕ taking values on ℐ
  (𝒲) and having the same law as ν =
  ℐ_ (ν) is a (weak) solution to
  equation (<ref>). For this reason in the following we call a map
  S : 𝒲→𝒲 such that T ∘ S =
  id_𝒲 μ^𝒜-almost surely a strong
  solution to equation (<ref>) and we call a measure ν on
  𝒲, such that T_ (ν) = μ^𝒜, a
  weak solution to equation (<ref>).



  Let T : (- Δ + 1)^- 1 (𝒲) → (- Δ + 1)^-
  1 (𝒲) be the map defined as T (w) = w +
  U (w) where w∈ (- Δ + 1)^- 1
  (𝒲)
  
    U (w) = ℐ (𝒜^2 [f (x) g (z) V'
         (w)]) .

  Then we have that ϕ (x, z, w) is a weak solution to equation if and
  only if T_∗ (ν_ϕ) = ℐ (μ^A) (where is the law of
  ϕ). Furthermore we have that if S is a strong solution to equation
  (<ref>) in the sense of Definition <ref> if and only
  if
  
    T (ℐ (S ((- Δ + m^2 + 𝔏) w))) = w

  for ℐ_* (μ^A)-almost every w∈ (- Δ +
  1)^- 1 (𝒲) (we implicitly use the fact that (- Δ + m^2 + 𝔏)
  (w) ∈𝒲 for ℐ (μ^A)-almost every w∈ (-
  Δ + 1)^- 1 (𝒲)).


In order to solve equation (<ref>) we need to introduce an
approximation. Let P_n be the orthogonal projection in L^2 (M) onto the
finite dimensional subspace generated by H_1, …, H_n. The restrictions
P_n |_𝒜 [L^2 (M)] and P_n
|_𝒜^1 / 2 [L^2 (M)] of P_n on 𝒜 [L^2 (M)] and
𝒜^1 / 2 [L^2 (M)] are the orthogonal projections on the subspace
generated by H_1, …, H_n too. This implies also that I ⊗ P_n
(in the following denoted also simply by P_n) is a continuous linear
operator on 𝒲.

Let ϕ_n be the solution to the following approximated equation

    (- Δ_x + 𝔏 + m^2) (ϕ_n) (x, z) + P_n 𝒜^2 [f g V' (P_n (ϕ_n))] (x, z) =   ξ^𝒜 (x, z) ,

where f g V' (P_n (ϕ_n)) is defined by equation (<ref>), and let U_n (w) = P_n (U (P_n (w))) and T_n (w) = w + U_n (w) be the maps,
analogous to U and T, related to equation (<ref>). Using
these objects we can define weak and strong solutions to
equation (<ref>) as in Definition <ref>.


Let us now study equation (<ref>). We introduce the following
function V_n : ℝ^n →ℝ

    V_n (y) = ∫_M g (z) V ( ∑_k = 1^n y^k H_k (z) ) 
       z.

Then, since P_n commutes with 𝔏 and 𝒜, being the
projection on a subset of common eigenfunctions of both 𝔏 and
𝒜, we have that ϕ_n is of the form

    ϕ_n (x, z) = ℐξ^𝒜 (x, z) +
       ϕ̅_n (x, z) = ℐξ^𝒜 (x, z) + ∑_k = 1^n
       ψ̅_n^k (x) H_k (z),

where ψ̅^k_n (x) solves the set of equations

    [                                                                                                                              0                                                                                                                              =                                                                                             (- Δ_x + m^2 + λ_k) (ψ̅^k_n) (x) +;                                                                                                                                                                                                                                                               + σ_k^2 f (x) ∫_M g (z) . V' ( ∑_j =
        1^n σ_j ℐ_λ_jξ^j (x) H_j (z) + . ∑_j =
        1^n ψ̅^j_n (x)  H_j (z) ) H_k (z)  z
        .;                                                                                                                                                                                                                                                             =                                            (- Δ_x + m^2 + λ_k) (ψ̅^k_n) (x) + f (x)
        σ_k^2 [∂_y^k (V_n) (ξ̃_n (x) + ψ̅_n)], ]

with  ℐ_λ_j = (- Δ_x + λ_j + m^2)^- 1,
ξ̃_n (x) = (σ_k ℐ_λ_k (ξ^k))_k = 1,
…, n∈ℝ^n, ∂_y^k is the partial derivatives with
respect the k-th variables y^k = ψ̅^k_n, and we used the fact that
𝒜 is self-adjoint in L^2 (M). If we denote by 𝒜_n
the n × n diagonal matrix such that 𝒜^_i j_n = σ_j
δ^i j, i, j = 1, …, n, and by 𝔏_n the n × n
matrix such that 𝔏_n^i j = λ_i δ^i j we can write
equation (<ref>) in the following way

    (- Δ_x + m^2 + 𝔏_n) (ψ_n) + f 𝒜^2_n ·∂ V_n (ψ_n) = 𝒜_n ·ξ_n,

where ψ_n = (ψ̅_n^k + ξ̃_n^k) |_k = 1,
…, n and ξ_n = (ξ^k) |_k = 1, …, n. Equation (<ref>) can be reformulated defined on the
abstract Wiener space (𝒲_n, H_n, μ̂_n),
where ℋ_n is the Cameron-Martin space

    ℋ_n  L^2 (ℝ^2 ; ℝ^n),

with the scalar product and norm given by ⟨ h, g ⟩ = ∑_i =
1^n 1/σ_i^2∫_ℝ^2 h^i (x) g^i (x)  x; the
Banach space 𝒲_n (in which ℋ_n is
densely embedded) is defined as

    𝒲_n  W^- 1 -, p_ℓ (ℝ^2 ;
       ℝ^n) ∩ (1 - Δ_x) (C^0_ℓ (ℝ^2 ;
       ℝ^n))

(where p ⩾ 1 is large enough), and μ̂_n is the law of the
noise ξ̂_n = (σ_1 ξ^1, …, σ_n ξ^n). On
𝒲_n we can introduce the corresponding maps Û_n
(ŵ_n) (x) = f (x)  𝒜^2_n ·∂ V_n
(ℐ_𝔏_n (ŵ_n) (x)) and T̂_n (ŵ_n) =
ŵ_n + Û_n (ŵ_n) where ℐ_𝔏_n is the
matrix valued operator defined as ℐ_𝔏_n^i j =
δ^i jℐ_λ_j and ŵ_n ∈W_n. Using
T̂_n we can define the concept of strong and weak solutions to
equation (<ref>).


  It is important to note that the relationship between ψ_n and the
  strong solution Ŝ_n (defined using the map T̂_n) is given by
  ψ_n (ŵ_n, x) =  ℐ_𝔏_n (Ŝ_n
  (ŵ_n)) (x), x ∈ℝ. Furthermore if ν̂_n is a
  weak solution to equation (<ref>) (i.e. such that T̂_n,
   (ν̂_n) = μ̂_n) we have that ν̂_n is the
  law of (- Δ_x + m^2 + 𝔏_n) (ψ_n).


In the following we shall denote by κ̂^n the probability law on
ℝ^n such that

    κ̂^n/ y = exp( - 4 π(
       1/2 | (m^2 +𝔏_n)^1 / 2 (𝒜_n^- 1·
       y) |^2 + V_n (y) ) )/Z_κ̂^n,

where Z_κ̂^n is a suitable constant and y is the
Lebesgue measure on ℝ^n. Finally we define Υ̃_f,
n as the random variable defined on 𝒲_n such that

    Υ̃_f, n (ŵ_n) : = exp( 4 ∫_ℝ^2
       f' (x) V_n (ℐ_𝔏_n (ŵ_n) (x))  x )

where ŵ_n ∈W_n. In the following we introduce the
function

    V^A_n (y) = V_n (σ_1 y^1, …, σ_n y^n) = ∫_M g (z) V
       ( A ( ∑_k = 1^n y^k H_k (z) ) )  z.


  If V^𝒜_n satisfies Hypothesis QC then
  there exists at least a weak solution ν̂^n to
  equation (<ref>) such that for any bounded measurable function
  F : ℝ^n →ℝ we have
  
    ∫_𝒲_n F ( ℐ_𝔏_nŵ_n (0)) Υ̃_f, n (w) ν̂_n (w) = Z_f,
        n∫_ℝ^n F (y) κ̂^n

  where
  
    Z_f, n : = ∫_𝒲_nΥ̃_f, n
        (ŵ_n) ν̂_n (ŵ_n) .




  The proof is given in <cit.> Theorem 1 for the case
  𝔏_n = 0 and 𝒜_n = I_n. The case of generic 𝒜_n can be
  obtained with a change of variables as it is described in Remark 5
  of <cit.>. The case with 𝔏_n a generic
  positive diagonal matrix is a trivial extension.


It is simple to prove that if the potential V satisfies Hypothesis QC the
potential V_n^𝒜 satisfies Hypothesis QC too, indeed the
following proposition holds.


  If V : ℝ→ℝ satisfies Hypothesis QC then
  V_n^𝒜 satisfies Hypothesis QC. Finally if V is strictly
  convex (i.e. it satisfies Hypothesis C) also V_n^𝒜 and V_n
  are convex.



  First of all we note that
  
    ∂_y^k V^𝒜_n (y) = ∫_M g (z) V' ( 
         𝒜( ∑_j = 1^n y^j H_j (z) ) ) 𝒜
         (H_k) (z)  z.

  This means that if n∈𝕊^n - 1 we have
  
    -n·∂ V_n^𝒜 (y + tn)     =     -
        ∑_k = 1^n n^k ∂_y^k (V_n^𝒜) (y + tn)
        
        =     - ∑_k = 1^n ∫_M n^k g (z) V' ( 𝒜∑_j =
        1^n y^j H_j (z) + 𝒜∑_j = 1^n n^j H_j (z) )
        𝒜 (H_k) (z)  z 
       ≲   ∫_M g (z) ℌ( 𝒜∑_j = 1^n
        y^j H_j (z) )  z 
       ≲   ℌ( | y | ∑_k = 1^+ ∞σ_k
         H_k 1_g (z) ≠ 0_∞)

  where we use that g has compact support, the fact that V satisfies
  Hypothesis QC and the fact that ℌ is increasing. Since
  
    ∑_k = 1^+ ∞σ_k  H_k 1_g (z) ≠
         0_∞⩽1/𝔴1_g (z)
         ≠ 0_∞∑_k = 1^+ ∞σ_k  H_k
         _∞, 𝔴 < + ∞,

  where 𝔴 : M →ℝ_+ is the weight function in
  Hypothesis H𝒜, and thus by Hypothesis H𝒜, the thesis
  is proved.


Using the weak solution ν̂_n to equation (<ref>), given
by Theorem <ref>, we are able to construct a weak solution to
equation (<ref>) satisfying the dimensional reduction principle.

A weak solution ν_n to equation (<ref>) is of the form
ν_n = ν̃_n ⊗μ_n^𝒜, where ν̃_n is
the law of

    (- Δ_x + m^2 + 𝔏) P_n (ϕ_n) (x, z) = ∑_k = 1^n (-
       Δ_x + m^2 + λ_k) (ψ^k_n) (x) H_k (z),

on the subspace Im (P_n) ⊂𝒲 and
μ_n^𝒜 is the law of

    Q_n (ϕ_n) = (- Δ_x + m^2 + 𝔏) (I - P_n) (ϕ_n) (x, z)
       = ∑_k = n + 1^+ ∞σ_k ξ^k (x) H_k (z),

which is the law of the Gaussian field on Im (Q_n) ⊂𝒲. Using the basis H_1, …, H_n we can identify the law
ν̃_n on Im (P_n) with a probability measure ν̂_n
on 𝒲_n. In this way it is evident that if the law
ν̂_n satisfies the dimensional reduction principle on
𝒲_n then the probability law ν_n = ν̂_n
⊗μ_n^𝒜 satisfies the dimensional reduction principle on
𝒲, since it is the tensor product of two probability laws
satisfying the dimensional reduction principle.

More precisely, we consider the following abstract Wiener space (𝕎,
ℍ, μ^𝒜, 𝔏) where

    𝕎    =    𝒜^1 / 2 (L^2 (M)), 
    ℍ    =     (𝔏 + m^2)^- 1 / 2 (𝒜 (L^2
      (M))),

(with 𝒜 and 𝔏 as in (<ref>)) and
𝕎 is equipped with its natural norm while on ℍ we define
the following scalar product

    ⟨ h_1, h_2 ⟩_ℍ = 4 π∫_M (𝔏 + m^2)^1
      / 2 (𝒜^- 1 (h_1)) (z) (𝔏 + m^2)^1 / 2
      (𝒜^- 1 (h_2)) (z)  z.

Thus μ^𝒜, 𝔏 is the centered Gaussian measure on
𝕎 with Cameron-Martin space given by ℍ. We now denote by
κ_n the measure on 𝕎 such that

    κ_n/μ^𝒜, 𝔏 (ω) =
      . exp( - 4 π∫_M g (z) V (P_n ( ω)  (z))  z ) / Z_κ_n.

where ω∈𝕎 and Z_κ_n is a renormalization
constant.


We introduce also the following random variables

    Υ_f, n (w) : = exp( 4 ∫_ℝ^2 × M f' (x) g
      (z) V (P_n (ℐ w) (x, z))  x  z )


    Υ_f (w) : = exp( 4 ∫_ℝ^2 × M f' (x) g (z) V
      ( (ℐ w) (x, z))  x  z )

where w ∈𝒲. If we use the previous identification of Im
(P_n) with 𝒲_n we have that

    Υ̃_f, n (P_n (w)) = Υ_f, n (w) .

Using the previous observations the next proposition (expressing the
dimensional reduction principle) trivially follows.


  If V satisfies Hypothesis QC then there exists
  a weak solution ν_n to equation (<ref>) such that for any
  bounded measurable function F : 𝕎→ℝ we have
  that
  
    ∫_𝒲 F (ℐ w (0, ·)) Υ_f, n (w) ν_n (w) = Z_f, n∫_𝕎 F (ω) κ_n
        (ω),

  with Z_f, n as in (<ref>).


The rest of the section concerns the generalization of
Proposition <ref> to solutions to
equation (<ref>). In particular we denote by κ the
probability measure on 𝕎 such that

    κ/μ^𝒜, 𝔏 (ω) =
      . exp( - 4 π∫_M g (z) V (ω (z))  z )
      / Z_κ. ,

where ω∈𝕎.


  Suppose that V satisfies Hypothesis QC then there
  exists at least a weak solution ν to equation (<ref>) such
  that for any bounded Borel measurable function F : 𝕎→ℝ we have
  
    ∫_𝒲 F ( ℐ w (0, ·)) Υ_f (w)
        ν (w) = Z_f ∫_𝕎 F (ω) κ (ω)
        ,

  where κ is defined as in (<ref>), Υ_f is given
  by equation (<ref>), and
  
    Z_f = ∫_𝒲Υ_f (w) ν (w) .




  Since κ_n converges weakly to κ, the right hand side of
  equation (<ref>) converges to the right hand side of
  equation (<ref>), as n → + ∞. 


In order to prove Theorem <ref> we prove the following lemmas. In
the space of functions C^0_ℓ (ℝ^2, ℝ^n) we denote by
·^𝒜_ℓ, k the following norm

    h ^𝒜_ℓ, k    =    sup_x ∈ℝ^2√(∑_j = 1^n σ_j^2 k (h^j (x))^2 r_ℓ (x)^2),

where h ∈ C^0 (ℝ^2, ℝ^n) and k, ℓ∈ℝ
(r_ℓ is the weight we introduced in (<ref>)).


  Let ψ̅_n be a solution to the
  equation (<ref>) and let ℓ, ℓ' ∈ℝ, then we
  have
  
    ψ̅_n ^𝒜_ℓ, - 1≲exp (αΞ_ℓ', n r_- ℓ' (x) 𝔴^- 1 (z)) 1_g (z)
        ≠ 0 f (x) r_ℓ (x) _∞

  
    [                                                              (- Δ_x + m^2 + 𝔏_n) (ψ̅_n)
          ^𝒜_ℓ, - 1; ≲exp (α (α' (Ξ_ℓ', n𝔴^- 1 (z) + ψ̅_n ^𝒜_ℓ', - 1)
          r_- ℓ' (x))) 1_g (z) ≠ 0 f (x) r_ℓ (x)
          _∞, ]

  where Ξ_ℓ', n =  P_n ℐξ^𝒜 (x, z)
  _ C^0_ℓ' (ℝ^2 ) ⊗_ϵ
  C^0_𝔴 (M), α, α' and the implicit constants do
  not depend on n.



  We write ψ̃_n = ( ψ̅_n^k/σ_k)_k
  = 1, …, n. The equation for ψ̃_n reads
  
    (- Δ_x + m^2 + λ_k) (ψ̃^k_n) (x) + σ_k f (x)
         ∫_M g (z) H_k (z) V' (P_n ℐξ^A (x, z) +
         ϕ̅_n (x, z))  z = 0

  where ϕ̅_n (x, z) is related to ψ̃_n by
  
    ϕ̅_n (x, z) = ∑_k = 1^n σ_k ψ̃^k_n (x) H_k
         (z) .

  Putting Ψ̃_n^2 (x) = (1 + θ | x |^2)^- ℓ∑_k = 1^n
  (ψ̃^k_n (x))^2, writing r_ℓ, θ (x) = (1 + θ | x
  |^2)^- ℓ, for some θ > 0, and denoting by x̅ the maximum
  of Ψ̃_n we have
  
    m^2 Ψ̃_n^2 (x̅)    ⩽    - 1/2Δ
        (Ψ̃_n^2) (x̅) + m^2 Ψ̃_n^2 (x̅) + r_ℓ,
        θ (x̅) ψ̃_n (x̅) ·𝔏_n
        (ψ̃_n (x̅)) 
       ⩽    - r_ℓ, θψ̃_n ·Δψ̃_n - r_ℓ, θ∑_k = 1^n | ∇ψ̃_n^k
        |^2 - ( - 2 | ∇ r_ℓ, θ |^2 + r_ℓ, θΔ r_ℓ, θ/2 r_ℓ, θ^2) Ψ̃^2_n
        (x̅) 
             + m^2 Ψ̃_n^2 (x̅) + r_ℓ, θψ̃_n (x̅) ·𝔏_n (ψ̃_n (x̅))
        
       ⩽    - f (x̅) r_ℓ, θ (x̅) ∫_M g (z)
        ϕ̅_n (x̅, z) V' (P_n ℐξ^A
        (x̅, z) + ϕ̅_n (x̅, z))  z 
             - ( - 2 | ∇ r_ℓ, θ |^2 + r_ℓ,
        θΔ r_ℓ, θ/2 r_ℓ, θ^2)
        Ψ̃^2_n (x̅),

  where we used that
  
    ∇[ ∑_k = 1^n (ψ̃^k_n (x̅))^2 ] = -
         ∇ r_ℓ, θ (x̅)/r_ℓ, θ (x̅)∑_k = 1^n (ψ̃^k_n (x̅))^2

  since x̅ is a stationary point for Ψ̃_n. From
  equation (<ref>) if, for any fixed ℓ, we choose a θ
  in such a way that
  
    | - 2 | ∇ r_ℓ, θ |^2 + r_ℓ, θΔ
         r_ℓ, θ/2 r_ℓ, θ^2| < m^2,

  we obtain
  
    Ψ̃_n^2 (x̅) ≲exp (αΞ_η', n
         r_- ℓ' (x) 𝔴^- 1 (z)) 1_g (z) ≠ 0 f
         (x) (r_ℓ, θ (x))^1/2_∞·
         (r_ℓ, θ (x))^1/2ϕ̅_n (x, z) 1_g
         (z) ≠ 0_∞,

  where all the constants do not depend on n. On the other hand
  
    (r_ℓ, θ (x))^1/2 | ϕ̅_n (x, z) |    ⩽   
        (r_ℓ, θ (x))^1/2∑_k = 1^n | ψ̅_n^k (x) |
        · | H_k (z) |
       ≲   1/𝔴 (z)1_g (z) ≠ 0_∞Ψ̃_n (x̅) √(∑_k = 1^∞σ_k  H_k (z) 𝔴 (z) _∞^2),

  where w (z) is the weight function in Hypothesis H𝒜, for all
  x ∈ℝ^2 and z ∈supp (g). Using
  Hypothesis H𝒜 we deduce from this that
  
    Ψ̃_n (x̅) ≲exp (αΞ_ℓ', n
         r_- ℓ' (x) 𝔴^- 1 (z)) 1_g (z) ≠ 0 f (x)
         (r_ℓ, θ (x))^1/2_∞ .

  Since ψ̅_n ^𝒜_ℓ, - 1≲_θΨ̃_n (x̅) inequality (<ref>) is proved.
  Inequality (<ref>) follows directly from
  inequality (<ref>) and equation (<ref>).



  Under the hypotheses and notations of
  Lemma <ref>, if ℓ < - 1, we have that there exists an
  increasing continuous function K_0 : ℝ_+ →ℝ_+
  such that
  
    (- Δ_x + m^2 + 𝔏) (ϕ̅_n) _ℋ⩽ K_0 (Ξ_ℓ, n),

  with Ξ_ℓ, n as in Lemma <ref>.



  The proof consists simply in noting that
  
    (- Δ_x + m^2 + 𝔏) (ϕ̅_n) _ℋ≲ (- Δ_x + m^2 + 𝔏_n) (ψ̅_n)
         ^𝒜_ℓ, - 1,

  indeed
  
    (- Δ_x + m^2 + 𝔏) (ϕ̅_n) _ℋ^2     =
           ∫_ℝ^2 × M [(- Δ_x + m^2 + 𝔏)
        (𝒜^- 1 (ϕ̅_n) (x, z))]^2  x  z
        =    ∑_k = 1^n 1/σ_k^2∫_ℝ^2 ((-
        Δ_x + m^2 + λ_k) (ψ̅_n) (x))^2  x
       ⩽    ( (- Δ_x + m^2 + 𝔏_n) (ψ̅_n)
        ^𝒜_ℓ, - 1)^2 ∫_ℝ^2 (1 + | x |^2)^ℓ x

  which is finite whenever ℓ < - 1.



  Since we can identify ξ^𝒜 with the identity map w
  ↦ξ^𝒜 (w) = w on 𝒲, we can identify Ξ_n, ℓ with a random
  variable defined as w ↦Ξ_n, ℓ (w) =  P_n (ℐξ^𝒜 (w)) (x, z) _ C^0_ℓ' (ℝ^2
  ) ⊗_ϵ C^0_𝔴 (M).


An important consequence of Corollary <ref> is the
following lemma.


  Let K ⊂𝒲 be a compact set, then
  𝔎 = ∪_n ∈ℕ T^- 1_n (K) is precompact in
  𝒲.



  We write 𝒯_n (ϕ̂, w) = ϕ̂ + U_n (ϕ̂ + w)
  . If w_n, y∈𝒲 is solution to the equation T_n  (w_n,
  y) = y (where y ∈ W) then ϕ̂_n, y = w_n, y - y will be a
  solution to the equation 𝒯_n (ϕ̂_n, y, y) = 0. This
  will implies that ϕ̅_n, y = ℐ (ϕ̂_n, y) is a
  solution to equation (<ref>) for the realization of the white
  noise ξ^𝒜 (y). By Corollary <ref> and
  Remark <ref>, we have that ϕ̂_n, y_ℋ⩽ K_0 (Ξ_ℓ, n (y)). On the other hand we have the estimate
  
    sup_n ∈ℕ (Ξ_ℓ, n (y)) = sup_n ∈ℕ
         P_n ℐξ^𝒜 (y) _ C^0_ℓ
         (ℝ^2 ) ⊗_ϵ C^0_𝔴 (M)≲sup_n ∈ℕ P_n ξ^𝒜 (y)
         _𝒲≲ξ^𝒜 (y) _𝒲,

  which implies that
  
    sup_n ∈ℕ, y ∈ Kϕ̂_n, y_ℋ⩽sup_n ∈ℕ, y ∈ K K_0 (Ξ_ℓ, n (y))
         ≲sup_y ∈ K K_0 (ξ^𝒜 (y) _𝒲) =
         C_K < + ∞ .

  Letting K̃ = { h ∈ℋ,  h _ℋ⩽
  C_K } we have that K̃ is compact in 𝒲 since the
  inclusion map i : ℋ→𝒲 is compact. This
  fact implies that 𝔎⊂ K + K̃ is precompact since
  it is contained in the sum of two compact sets.



  The family (ν_n)_n of measures such that T_n,
   (ν_n) = μ^𝒜 is tight.



  Let K̃ be a compact set such that μ^𝒜 (K̃)
  ⩾ 1 - ϵ for a fixed 0 < ϵ < 1, then, by
  Lemma <ref>, 𝔎∪_n ∈ℕ T^- 1_n (K̃) is a compact set in 𝒲. This
  implies
  
    ν_n (𝔎) ⩾ν_n (∪_i T^- 1_i (K̃))
         ⩾ν_n (T^- 1_n (K̃)) ⩾μ^𝒜
         (K̃) ⩾ 1 - ϵ,

  for any n ∈ℕ. Thus the sequence ν_n is tight and the lemma
  is proven. 



  Suppose that (ν_n)_n is a sequence of weak
  solutions to equation (<ref>) weakly converging to ν, then
  ν is a solution to (<ref>), i.e. T_ (ν) =
  μ^𝒜.



  Proving the claim is equivalent to prove that for any function h :
  𝒲→ℝ which is bounded with continuous and
  bounded Fréchet derivative we have ∫ h ∘ T ν = ∫ h
  μ^𝒜. In order to prove this we note that
  
    h ∘ T (w) - h ∘ T_n (w) _𝒲⩽ h
         _C^1 (𝒲)· U (w) - P_n U (P_n (w)) _𝒲
         .

  On the other hand we have
  
    U (w) - P_n U (P_n (w)) _𝒲⩽√(∑_k =
         n^∞σ_k)· U (w) _ℋ + sup_y ∈ [w,
         P_n (w)]∇ U (y) _ℒ (𝒲) P_n (w) - w
         _𝒲 .

  Finally we observe that
  
    P_n (w) - w _𝒲 =  (I - P_n) (T_n (w)) _𝒲
         .

  Let, for any fixed ε > 0, K_ϵ⊂𝒲 be
  a compact set such that ν_n (𝒲\ K_ϵ), ν
  (𝒲\ K_ϵ) < ϵ, then we have
  
    | ∫_𝒲 h ∘ T ν - ∫_𝒲 h
        μ^A |    ⩽   | ∫_𝒲 h ∘ T
        ν - ∫_𝒲 h ∘ T ν_n | + |
        ∫_𝒲 (h ∘ T - h ∘ T_n) ν_n |
        
             + | ∫_𝒲 h ∘ T_n ν_n -
        ∫_𝒲 h μ^A | 
       ⩽   | ∫_𝒲 h ∘ T ν -
        ∫_𝒲 h ∘ T ν_n | + ϵ h _C^0
        (𝒲)
             + | ∫_K_ϵ (h ∘ T - h ∘ T_n) ν_n | .

  But
  
    | ∫_K_ϵ (h ∘ T - h ∘ T_n) ν_n |
           ≲_h    √(∑_k = n^∞σ_k)·sup_w ∈
        K_ϵ U (w) _ℋ + sup_w ∈ K_ϵ∇ U (w) _ℒ (𝒲)
            ×∫_𝒲 (I - P_n) (T_n (w))
        _𝒲ν_n .

  On the other hand
  
    ∫_𝒲 (I - P_n) (T_n (w)) _𝒲ν_n =
        ∫_𝒲 (I - P_n) (w) _𝒲μ^𝒜≲∑_k = n^∞σ_k
        ,

  where we used that 𝔼 [ (- Δ_x + 1)^- 1 (ξ^k)
  ^2_C^0_ℓ (ℝ^2) ∩ W^1 -, p_ℓ (ℝ^2)] is
  equal to a finite constant independent on k. Since ∑σ_k is
  convergent, we have that the right hand side of (<ref>)
  converges to 0 as n → + ∞. Since U (w), ∇ U (w) are
  continuous with respect to w, this implies that the right hand side of
  equation (<ref>), converges to 0 as n → + ∞.
  Exploiting this fact, the fact that ν_n weakly converges to ν and
  equation (<ref>) we obtain that | ∫_𝒲 h
  ∘ T ν - ∫_𝒲 h μ^𝒜|
  ⩽ϵ h _C^0 (𝒲). Since ϵ > 0 is
  arbitrary the lemma is proved.


Proof of Theorem <ref>
  Thanks to Lemma <ref> we have that the sequence of weak
  solutions ν_n, satisfying the dimensional reduction principle (whose
  existence is proven by Proposition <ref>) converges
  weakly, as n → + ∞, to a solution ν of the
  equation (<ref>). If we are able to prove that Υ_f, nν_n →Υ_f  ν weakly, then the theorem would follow.
  
  First of all we note that Υ_f, n (w) = Υ_f (P_n (w)) and
  that Υ_f is a bounded Fréchet C^1 (𝒲) function. We
  want to prove that ∇Υ_f (w) is bounded when w is in a
  bounded subset of 𝒲. We have that
  
    ∇Υ_f (w) [h] = - Υ_f (w) ·∫_ℝ^2
         × M f' (x) g (z) V' (ℐ (w) (x, z)) (ℐ h) (x, z)
          x  z.

  In particular we have that
  
    ∇Υ_f (w) _𝒲^≲∫_ℝ^2 × M | f' (x) g (z) | exp (αℐ
         (w) _C^0_ℓ (ℝ^2) ⊗_ϵ C^0_𝔴
         (M) r_- ℓ (x) 𝔴^- 1 (z))  x  z,

  for a suitable positive constant α. On the other hand the linear map
  ℐ : 𝒲→ C^0_ℓ (ℝ^2)
  ⊗_ϵ C^0_𝔴 (M) is continuous, which implies that
  if B is a bounded subset of 𝒲 given by sup_w ∈ Bℐ (w) _C^0_ℓ (ℝ^2) ⊗_ϵ
  C^0_𝔴 (M) < + ∞. This observation proves that sup_w
  ∈ B∇Υ_f (w) _𝒲^ < + ∞ for
  any bounded set B ⊂𝒲.
  
  Let for any given ϵ > 0, K_ϵ⊂𝒲 be a
  compact set such that ν_n (𝒲\ K_ϵ), ν
  (𝒲\ K_ϵ) < ϵ, then there exists a ball
  B_ϵ⊂𝒲 such that K_ϵ∪ (∪_n ∈ℕ P_n (K_ϵ)) ⊂ B_ϵ. Let F be a
  continuous and bounded function on 𝒲, then
  
    | ∫_𝒲 F (w) Υ_f, nν_n -
        ∫_𝒲 F (w) Υ_f ν|    ⩽   | ∫_𝒲 F (w) (Υ_f (P_n (w)) - Υ_f (w))
        ν_n | 
             + | ∫_𝒲 F (w) Υ_f ν_n -
        ∫_𝒲 F (w) Υ_f ν| 
       ≲   ϵ + ∇Υ_f (w) _C^1 (B_ϵ)∫_𝒲 P_n (w) - w _𝒲ν_n
        
             + | ∫_𝒲 F (w) Υ_f ν_n -
        ∫_𝒲 F (w) Υ_f ν| 
       ≲   ϵ + ∇Υ_f (w) _C^1 (B_ϵ)∫_𝒲 P_n (T_n (w)) - T_n (w) _𝒲ν_n 
             + | ∫_𝒲 F (w) Υ_f ν_n -
        ∫_𝒲 F (w) Υ_f ν| →ϵ,

  as n →∞, where the constants implied in the symbol
  ≲ do not depend on ϵ. For this reason since ϵ is
  arbitrary positive, ∫_𝒲 F (w) Υ_f, nν_n
  →∫_𝒲 F (w) Υ_f ν and the theorem
  is proven.




 §.§ Dimensional reduction in the full
space


In this section we consider the following equation

    (- Δ_x - Δ_z + m^2) (ϕ) (x, z) + 𝒜^2 [f g
      V'(ϕ) ] (x, z) = ξ^𝒜 (x, z) ,

where x ∈ℝ^2 and z ∈ M (see equation (<ref>) for the notation fgV'(ϕ):=fg∂ V(ϕ)). We consider the following
hypotheses on 𝔏 and 𝒜 (in addition to the previous
Hf and Hg on the cut-offs):


  
  Hypothesis H𝔏1 𝔏 = - Δ_z and M
  =ℝ^d (for any fixed integer d ⩾ 0).
  
  
  Hypothesis H𝒜1 𝒜 (h (z')) = 𝔞 h where 𝔞 (z) is a 𝒞^1 + d / 2 +
  (ℝ^2) Hölder continuous function with compact support.
  
  
  Hypothesis Hξ 1 The noise ξ^𝒜 is such that if
  h_1, h_2 ∈ C^∞_0 (ℝ^2 ×ℝ^d) we have
  
    𝔼 [⟨ h_1, ξ^𝒜⟩⟨ h_2,
         ξ^𝒜⟩] = ∫_ℝ^2 ×ℝ^d (I
         ⊗𝒜) (h_1) (x, z) (I ⊗𝒜) (h_2) (x, z)
          x  z.

  
  
In this subsection we assume Hypotheses H𝔏1, H𝒜1
and Hξ1 instead of the corresponding assumptions
H𝔏, H𝒜 and Hξ respectively we had before. The
assumptions Hf and Hg on the space cut-offs however remain.


We define the abstract Wiener space (𝒲, ℋ,
μ^𝒜) relative to equation (<ref>) as follows

    ℋ    =     L^2 (ℝ^2) ⊗_H 𝒜 (L^2
      (ℝ^d)) 
    𝒲    =     (- Δ_x + 1) (C^0_ℓ (ℝ^2) ∩ W^1 -,
      p_ℓ (ℝ^2)) ⊗_ϵ C^0_ℓ' (ℝ^d)

where we suppose ℓ, ℓ' > 0. The maps T and U, and so the concept
of weak and strong solution to equation (<ref>), are defined
in the same way as in the previous section. We shall prove the following
theorem.


  Suppose that V satisfies Hypothesis QC (in
  Section <ref>), then there exists a weak solution ν to
  equation (<ref>) such that for any bounded Borel measurable
  function F : 𝕎→ℝ
  
    ∫_𝒲 F (ℐ w (0)) Υ_f (w) ν (w) = Z_f ∫_𝕎 F (ω) κ (ω),

  as in (<ref>).


It is clear that the conditions H𝔏1, H𝒜1 on
𝔏, M, 𝒜 are incompatible with the previous
Hypotheses H𝔏 and H𝒜. Thus
Theorem <ref> does not follow directly from
Theorem <ref>. The rest of this subsection will focus on the
proof of Theorem <ref>.


In order to achieve the proof we first replace M by the manifolds M_R
=𝕋^d_R, i.e. M_R is a torus of radius R > 0. If h is a
real-valued function on M =ℝ^d with sufficient decay at infinity we
can easily define a function h_R on M_R in the following way

    h_R (z_R) = ∑_j ∈ℤ^d h (z_R + R j)

where z_R ∈ [- R / 2, R / 2]^d. There is also a standard way of defining a
function h̃_R on M_R given one on M, that is the following

    h̃_R (z_R) = h (z_R) .

If the function h has support contained in [- R / 2, R / 2]^d then
h̃_R = h_R.

On the other hand if we consider a function k : M_R →ℝ
we can associated with it a function k^p defined on all of M in the
following way

    k^p (z) = k ( z - R ⌊z + R/2/R⌋),

z ∈ M =ℝ^d. In general we have that (k^p)_R = k,
and so if k has compact support contained in (- R / 2, R / 2)^d we have k
= (k^p)_R. Furthermore if 𝔟 is a function with compact support
on ℝ^d and k is a function on M_R we have (𝔟_R
 k)^p = 𝔟 k^p. Furthermore it is important to
note that if 𝔊_R is the Green function associated with the
operator (- Δ_x - Δ_z + m^2)^- 1 on ℝ^2 × M_R
and 𝒢 is the Green function of the same operator on ℝ^2
× M we have

    𝒢_R = 𝔊_R

We define the operator 𝒜_R on L^2 (M_R) by 𝒜_R (k) =
𝔞_R  k. In this way we can define the abstract Wiener
space (𝒲_R, ℋ_R, μ^𝒜_R_R) as follows

    ℋ_R     =     L^2 (ℝ^2) ⊗_H 𝒜_R (L^2 (M_R))
      
    𝒲_R     =     (- Δ_x + 1) (C^0_ℓ (ℝ^2) ∩ W^1 -,
      p_ℓ (ℝ^2)) ⊗_ϵ𝒜_R^1 / 2 (L^2
      (M_R))

and μ^𝒜_R_R is the law of the noise ξ^𝒜_R on
𝒲_R with covariance 𝒜_R.

Using the map (·)^p we can extend the noise ξ^𝒜_R,
defined on M_R, to the noise ξ^𝒜, p_R defined on all of M.
This means that the law μ^𝒜_R_R of ξ^𝒜_R on
𝒲_R, thanks to the map (·)^p, induces a Gaussian measure
μ^𝒜_R, p_R on 𝒲, which is the underling
probability, measure of the noise ξ^𝒜, p_R. It is simple to
prove that μ^𝒜_R, p_R weakly converges to μ^𝒜
when R → + ∞ and the support of 𝔞 is compact.

Let U_R : 𝒲_R →ℋ_R be the function defined
by

    U_R (w_R) : = f (x) 𝒜_R (g_R (z_R) V' (𝔊_R 
       w_R)) = f (x) 𝒜_R (g_R (z_R) V' (𝒢_R  w_R)) .

We set T_R (w_R) = w_R + U_R (w_R). The map U_R induces a map U_R^p
on 𝒲 in the following way

    U_R^p (w) = f (x) 𝒜_R (g_R^p (z_R) V' (𝒢
       (w̃_R)^p)),

and T^p_R (w) = w + U^p_R (w). Let ν_R be a probability law on
𝒲_R such that T_R,  (ν_R) = μ^𝒜_R_R,
then it induces a probability law ν^p_R on 𝒲 such that
T^p_R, ∗ (ν_R^p) = μ^𝒜_R, p_R.


  Let R_n ∈ℝ_+ be a sequence such that
  R_n → + ∞, then the sequence ν_R_n^p is tight on
  𝒲 and if ν_R_n^p →ν, as n →∞ and R_n →∞, then T_ (ν) =
  μ^𝒜.



  We note that the support of ν_R_n^p is contained in the set of R_n
  periodic distributions contained in 𝒲 and furthermore the map
  T_R_n^p sends R_n periodic distributions into R_n periodic
  distributions.
  
  Using the methods of Lemma <ref> and
  Corollary <ref>, it is simple to prove that if y_n ∈𝒲 is an R_n periodic distribution and if w_y_n∈𝒲 is an R_n periodic distribution such that w_y_n∈ T^p,
  - 1_R_n (y_n) then
  
    w_y_n - y_n _ℋ⩽ K_f, g (Ξ̅_ℓ,
        ℓ' (y_n)),

  where
  
    Ξ̅_ℓ, ℓ' (y_n) = sup_(x, z) ∈ℝ^2 × M
         | 𝒢 y_n (x, z) | r_ℓ (x) r_ℓ' (z),

  and K_f, g is a positive increasing continuous function depending only
  on the functions f and g. It is simple to prove that the map
  Ξ̅_ℓ, ℓ' is continuous on 𝒲. Using the fact
  that, as n →∞ and R_n →∞,
  μ^𝒜_R, p_R_n converges to μ^𝒜 weakly and
  so the sequence μ^𝒜_R, p_R_n is tight, we can use the
  bound (<ref>) and the same methods of
  Lemma <ref> to prove that ν_R_n^p is tight.
  
  Suppose that ν_R_n^p weakly converges to ν, we want to prove that
  T_ (ν) = μ^𝒜. Let F_R_n be a function of
  the form F_R_n (w) = G (⟨ h_1, w ⟩, …, ⟨ h_r,
  w ⟩), where G : ℝ^r →ℝ is  a
  continuous and bounded function and h_1, …, h_r are smooth functions
  with support in (- R / 2 + 𝔯, R / 2 - 𝔯)^n and
  𝔯 = diam (supp (𝔞)). For this kind of
  function we have that F_R_n∘ T_R_k^p = F_R_n∘ T for k
  ⩾ n. From this observation we get
  
    ∫ F_R_n∘ T ν = lim_k ∫ F_R_n∘ T ν_R_k^p = lim_k ∫ F_R_n∘ T_R_k^p ν_R_k^p =
         lim_k ∫ F_R_nμ^𝒜_R_k, p_R_k = ∫
         F_R_nμ^𝒜,

  where the limit is taken for k →∞ and R_k →∞. Since the functions of the form F_R_n, for n ∈ℕ,
  generate the space of all 𝒲 Borel measurable functions the lemma
  is proved.


Proof of Theorem <ref>
  By Theorem <ref> there exists a probability law ν_R_n on
  𝒲_R_n such that T_R_n,  (ν_R_n) =
  μ^𝒜_R_n_R_n. On the other hand this implies that
  T^p_R_n (ν_R_n^p) = μ^𝒜_R_n, p_R_n and so, by
  Lemma <ref>, there exists at least one probability measure
  ν on 𝒲 such that T_ (ν) = μ^𝒜
  and ν^p_R_n→ν weakly, as n →∞.
  
  Then, using the notations of the proof of Lemma <ref>, we
  have that for any bounded continuous function F_R_n : 𝕎→ℝ
  
    ∫_𝒲 F_R_n (𝒢 w (0, z)) Υ_f (w)
        ν^p_R_k (w) = ∫_𝕎 F_R_n (ω) κ_R_k^p (ω),

  where we used that
  
    ∫_𝒲 F_R_n (𝒢_R_k w_R_k (0, z))
         Υ_f (w_R_k) ν_R_k (w_R_k) = ∫_𝒲
         F_R_n (𝒢 w (0, z)) Υ_f (w) ν^p_R_k (w),

  for k ⩾ n and a similar equality for κ_R_k. Since
  F_R_k (𝒢·) and Υ_f are continuous on
  𝒲, the left hand side of (<ref>), converges to
  ∫_𝒲 F_R_n (𝒢 w (0, z)) Υ_f (w)
  ν (w) as k → + ∞.
  
  Furthermore, since
  
    . κ_R_k^p/μ^𝒜_R_k, p =
         Z^- 1_κ_R_k^p. . exp( - 4 π∫_ℝ^2 g (z) V (ω (z))  z ) .

  and since μ^𝒜_R_k, p_R_k weakly converges to
  μ^𝒜, 𝔏, we have that κ^p_R_k weakly
  converges to κ, as k, R_k →∞. This proves that the
  right hand side of (<ref>) converges to ∫_𝕎
  F_R_n (ω) κ (ω), as k, R_k →∞.
  Since the functions of the form F_R_n, for n ∈ℕ, generate
  the space of 𝕎 measurable functions, the theorem is proved.




 §.§ Cut-off removal with convex
potential


Hereafter we denote by ω_β (x) the function

    ω_β (x) exp (- β√((1 + | x |^2))),

β > 0, x ∈ℝ^2, and introduce the space
𝒲_β in the following way

    𝒲_β : = (- Δ + 1) C^0_expβ (ℝ^2)
       ⊗_ϵ𝒜^1 / 2 (L^2 (M))

where C^0_expβ is the space of continuous functions with respect to
the weighted L^∞ norm

    g _∞, expβsup_x ∈ℝ^2 |
       ω_β (x) g (x) |,

and M as before in Section <ref>.

In this section we want to prove the following theorem.


  Suppose that V is a convex function, and suppose
  that 𝒜, and 𝔏 satisfy Hypotheses H𝒜,
  H𝔏 and Hξ (or Hypotheses H𝒜1, H𝔏1
  and Hξ1) then there exists a unique strong solution ϕ (x, z) to
  equation (<ref>) (or to equation (<ref>)) with f
  ≡ 1 taking values on 𝒲_β (for a 0 < β⩽β_0 which depends only on m^2) such that for any 𝕎
  measurable bounded function F we have
  
    𝔼 [F (ϕ (0, z))] = ∫_𝕎 F (ω) κ
        (ω)

  (κ as in equation (<ref>)).


The proof is very similar to those of Theorem <ref> and
Theorem <ref>. For this reason we report here only the main
differences. First of all we need a replacement for
Proposition <ref>.


  Let V (and so V_n) be a convex function,
  then under assumptions H𝔏, H𝒜 and Hξ  there
  exists a unique strong solution ϕ_n to equation (<ref>)
  such that for all 𝕎 measurable bounded continuous function F we
  have
  
    𝔼 [F (ϕ_n (0, z))] = ∫_𝕎 F (ω) κ_n (ω)

  where κ_n is given by expression (<ref>).



  The proof is given in <cit.>, Theorem 2 in the case
  𝔏_n = 0. The case considered here is a trivial extension. 


In order to pass from equation (<ref>) to
equation (<ref>) we need a generalization of
Lemma <ref>. We denote by ·_expβ,
ℓ^𝒜 respectively ·^𝒜_U, k the
following norms

    [                                               h ^𝒜_expβ, k                                                            sup_x ∈ℝ^2√(∑_j = 1^n σ_j^2 k (h^j (x))^2
         ω_β (x)^2),;                                                  h ^𝒜_U, k                                                                      sup_x ∈ U√(∑_j =
         1^n σ_j^2 k (h^j (x))^2) . ]




  There exists a number β_0 > 0 (depending on
  m^2) such that for any 0 < β⩽β_0, and for any open
  bounded set U ⊂ℝ^2 and under
  Hypotheses C, Hg, H𝒜, H𝔏, Hξ and f ≡ 1
  we have
  
    ψ̅_n ^𝒜_expβ, - 1   ≲   exp (αΞ_ℓ', n r_- ℓ' (x) 𝔴^- 1 (z))
        1_g (z) ≠ 0ω_β (x) _∞
     (- Δ_x + m^2 + 𝔏_n) (ψ̅_n) ^𝒜_U,
        - 1   ≲   exp (α'
        (Ξ_ℓ', n r_- ℓ' (x) 𝔴^- 1 (z)  + 
           + ψ̅_n ^𝒜_expβ, - 1) ω_- β (x))
        _C^0 (U_g)

  where U_g = U ×{ g (z) ≠ 0 }⊂ℝ^2
  × M, uniformly in n (where Ξ_ℓ, n is defined as in Lemma
  <ref>).



  The proof is verbatim the same as for Lemma <ref> where we
  replace the function r_θ, ℓ (x) = (1 + θ | x |^2)^- ℓ
  by the function ω_β, defined by (<ref>),
  and we use the fact for β small enough we have
  
    | - 2 | ∇ω_β |^2 + ω_βΔω_β/2 ω_β^2| < m^2 .



The inequality (<ref>) implies that, for any bounded open
subset U of ℝ^2 we have

    (- Δ_x + m^2 + 𝔏) (ϕ̅_n) _ℋ_U⩽ K_U (Ξ_ℓ, n)

where ·_ℋ_U is the natural norm of the Hilbert space
ℋ_U = L^2 (U) ⊗_H 𝒜 (L^2 (M)), and K_U is a
positive increasing continuous function depending only on U.
Inequality (<ref>) guarantees us enough compactness to
generalize  Lemma <ref>, Lemma <ref> and
Lemma <ref> in order to prove the existence of a weak solution
to equation (<ref>) satisfying the dimensional reduction principle
when f ≡ 1 and under Hypotheses C, Hg, H𝒜,
H𝔏, Hξ. We can generalize the described result under
Hypotheses C, Hg, H𝒜1, H𝔏1, Hξ1 using a similar
strategy.

It remains to prove the uniqueness of the solution to
equation (<ref>) (or (<ref>)) when V is convex.
This can be done using the methods of Lemma <ref> and
Lemma <ref>. We give here only a sketch of the proof.

Let ϕ_1 and ϕ_2 be two strong solutions to
equation (<ref>). It is simple to prove that ϕ_1 (x, ·) -
ϕ_2 (x, ·) ∈𝒜 (L^2 (M)), so the following function is
well defined

    Ψ_β (x)^2 = ω_2 β (x) ∫_M (𝒜^- 1 (ϕ_1
       (x, z') - ϕ_2 (x, z')))^2  z,

where ω_2 β is defined as in equation
(<ref>). The function Ψ^2_β belongs to C^2
(ℝ^2)  and goes to zero at infinity. This means that the maximum is
realized in at least one point x̅∈ℝ^2 . Making some
computations similar to the ones of Lemma <ref> we obtain

    [                                                                            m^2 Ψ_β (x̅)^2                                                                                         ⩽ - ∫_M g (z) (ϕ_1
         (x̅, z) - ϕ_2 (x̅, z))(V' (ϕ_1 (x̅, z)) - V'
         (ϕ_2 (x̅, z)))  z;                                                                                                                                                                                                                  - ( - 2 | ∇ω_β |^2 +
         ω_βΔω_β/2 ω_β^2)
         Ψ_β (x̅)^2 . ]

On the other hand there exists a point θ_x̅, z∈ [ϕ_1
(x̅, z), ϕ_2 (x̅, z)] such that

    V' (ϕ_1 (x̅, z)) - V' (ϕ_2 (x̅, z)) ⩽ V”
       (θ_x̅, z) (ϕ_1 (x̅, z) - ϕ_2 (x̅, z)) .

Choosing β > 0 small enough we obtain

    sup_x ∈ℝ^2Ψ_β (x)^2 ≲ - ∫_M g (z) V”
       (θ_x̅, z) (ϕ_1 (x̅, z) - ϕ_2 (x̅, z))^2
        z ⩽ 0,

since V is convex. This implies that Ψ_β (x) ≡ 0 and so that
ϕ_1 (x, z) = ϕ_2 (x, z).



§ THE EXPONENTIAL INTERACTION ON
ℝ^2


In this section we want to consider the following elliptic SPDE

    (- Δ_x - Δ_z + m^2) (ϕ) + g (z) αexp (αϕ -
      ∞) = ξ

where x ∈ℝ^2, z ∈ M =ℝ^2, ξ = ξ (x, z) is a
standard Gaussian white noise on ℝ^4, | α | < 4 √(2)π, m > 0, g : ℝ^2 →ℝ is a non-negative
smooth function with compact support, and where - ∞ means that the
equation should be properly renormalized. In order to give a meaning to the
previous equation we formally subtract from the solution ϕ the solution
to the linear equation (i.e. equation (<ref>) with g = 0)
which means that we consider the equation for the unknown ϕ̅:

    (- Δ_x - Δ_z + m^2) (ϕ̅) + g (z) αexp (αϕ̅) η (x, z) = 0 ,

where η (x, z) = exp^♢ (αℐξ) is a
renormalized version of the distribution exp (αℐξ -
∞), where exp^♢ denotes the Wick exponential of the Gaussian
distribution ℐξ. Hereafter we denote in general by B^s_p, q,
ℓ (ℝ^d + 2) the weighted Besov space of indices 1 ⩽ p
⩽∞ and 1 ⩽ q ⩽∞ and weight given by
r̅_ℓ (x, z) = ( √(| x |^2 + | z |^2 + 1))^-
ℓ, where ℓ∈ℝ (see <cit.>). It is well
known that ℐξ∈ B^- δ_p, p, ℓ for any 1
⩽ p ⩽∞, δ > 0 and ℓ > 0 (see
e.g. <cit.>).


In the following we shall take d = 2 and give a rigorous meaning to
equation (<ref>) (and so to
equation (<ref>)) when the exponent | α | < α_max (see equation (<ref>) for the definition α_max) and, when αϕ̅⩽ 0, prove that there exists only
one solution to equation (<ref>).

Furthermore we want to prove that dimensional reduction holds for the unique
solution to equation (<ref>), namely that, if we consider
the measure κ_g given by

    κ_g/μ (ω) = exp( - 4 π∫_ℝ^2 g (z) exp^♢ (αω) ( z)
      ),

where ω∈ B^- δ_p, p, ℓ (ℝ^2) and where μ is
the law of (- Δ_z + m^2)^- 1 / 2 (ξ) on B^- δ_p, p, ℓ
(ℝ^2), we have

    𝔼 [F (ϕ (0, ·))] = ∫ F (ω) κ_g (ω)
       ,

for any bounded measurable function F defined on B^- δ_p, p, ℓ
(ℝ^2).

In order to prove the existence and uniqueness of the solution ϕ̅ to
equation (<ref>) and dimensional
reduction (<ref>) for the random field ϕ =
ϕ̅ + ℐξ, we need to introduce the following two
approximate equations

    (- Δ_x - Δ_z + m^2) (ϕ_ϵ) +
      𝔞_ϵ^ 2 [g (z) αexp (αϕ_ϵ - C_ϵ)] = 𝔞_ϵξ   
        (- Δ_x - Δ_z + m^2) (ϕ̅_ϵ) +
      𝔞_ϵ^ 2 [g (z) αexp (αϕ̅_ϵ) η_ϵ (x, z)] = 0

where 𝔞 is a positive function satisfying
Hypothesis H𝒜 1 (see Section <ref>) and
C_ϵ : = α^2/2𝔼 [ℐ
(𝔞_ϵξ)], ϕ̅_ϵ : =
ϕ_ϵ - ℐ (𝔞_ϵξ), and
η_ε is the positive measure defined as

    η_ε ( x,  z) exp^♢ (αℐ (𝔞_ϵξ))  x  z =
       exp (αℐ (𝔞_ϵξ) -
       C_ε)  x  z.

The constant C_ϵ is chosen in such a way that η_ϵ→η in B^s_p, p, ℓ for suitable s < 0, 1 < p ⩽
2 and ℓ > 0 (see Section <ref>).

For equations (<ref>) and (<ref>) we have
the following fundamental dimensional reduction result.


  Equation (<ref>) admits
  a unique solution in C^0_ℓ (ℝ^4) for ℓ big enough.
  Furthermore we have that, for any ϵ > 0 and any bounded continuous
  function F
  
    𝔼 [F (ϕ_ϵ (0, z))] = ∫_𝒞^0 -_ℓ
        (ℝ^2) F (ω) κ_ϵ (ω)
        ,

  for all z ∈ℝ^2, where
  
    κ_ϵ/μ_ϵ = exp( - 4
         π∫_ℝ^2 g (z) exp (αω (z) - α
         C_ϵ)  z )

  and μ_ϵ is the Gaussian measure on 𝒞^0 -_ℓ
  with covariance 𝔼 [ω (z) ω (z')] =
  𝔞^ 2_ϵ𝒢_z (z - z'),
  where 𝒢_z is the Green function of the operator (- Δ_z +
  m^2). Finally the unique solution to equation (<ref>)
  satisfies αϕ̅_ε⩽ 0.



  The proof is an application of Theorem <ref> to
  equation (<ref>) using the fact that exp (α y -
  C_ϵ) satisfies Hypothesis C in Section <ref>.
  
  The fact that αϕ̅_ε⩽ 0 follows from an
  application of the maximum principle to the function ẑ↦ϕ̅_ϵ (ẑ)  r̅_ℓ (λẑ) where
  ẑ = (x, z) ∈ℝ^4 and λ > 0 small enough.


In order to prove existence, uniqueness and the reduction principle for
equation (<ref>) we have now to study the behavior of the
regularized noises η_ϵ and their convergence to η in the
Besov spaces of the form B^s_p, p, ℓ, as ε→ 0.
Once we have established this convergence, we can give a meaning to
equation (<ref>) and we are able to prove that there exists
a subsequence of ϵ_n → 0 such that ϕ_ϵ_n→ϕ in probability in B^- δ_p, p, ℓ, where ϕ
solves (<ref>). This fact will permit us to prove
equality (<ref>).



 §.§ Probabilistic analysis


In this subsection we propose an analysis of the regularity of the noise
η and η_ϵ, for ϵ > 0. First of all we note that

    η_ϵ    =    ∑_k = 0^∞α^k/k! (ℐξ_ϵ)^♢ k,  
    η    =    ∑_k = 0^∞α^k/k! (ℐξ)^♢ k,

where I is the integral operator defined in Section <ref>,

    (ℐξ_ϵ)^♢ k = ℐξ_ϵ♢…♢ℐξ_ϵ_k
         times ,

the symbol ♢ denotes the Wick product, ξ_ϵ =
𝔞_ϵξ and ℐξ_ϵ is
defined correspondingly. The previous expressions are well defined as L^2
(μ) convergent series for | α | < 4 √(2)π. Indeed we have
that for any  smooth function g exponentially decaying at infinity 
    𝔼 [| ⟨η_ϵ, g ⟩ |^2] = ∫_ℝ^8
      g (ẑ) g (ẑ') exp (α^2 𝒢_ϵ (ẑ -
      ẑ')) ẑẑ'

where, hereafter, we write 𝒢_ϵ = 𝔞^
2_ϵ𝒢, where 𝒢 is the Green
function associated with the operator (- Δ_ẑ + m^2)^- 2,
ẑ = (x, z) ∈ℝ^4. It is well known (see Proposition A.1
of <cit.>) that for ẑ∈ℝ^4 such that |
ẑ | ⩽ 1 there exists a constant C_1 > 0 for which the
following inequality holds

    𝒢 (ẑ) ⩽ - 2/(4 π)^2log_+ (| ẑ |) +
      C_1  .

Furthermore for | ẑ | ⩾ 1 there exists two constants C_2, C_3
> 0 for which we get

    𝒢 (ẑ) ⩽ C_2 exp (- C_3 | ẑ |)
       .

An easy consequence of the inequalities (<ref>)
and (<ref>) are the following inequalities

    𝔞_ϵ^ 2𝒢 (ẑ)
      ⩽ - 2/(4 π)^2log_+ (| ẑ |) + C_4,

when 4 diam (supp (𝔞)) ϵ⩽ | ẑ
| ⩽ 1 and for some constant C_4 > 0; 
    𝔞_ϵ^ 2𝒢 (ẑ)
      ⩽ - 2/(4 π)^2log (ϵ) + C_5,

when | ẑ | < 2 diam (supp (𝔞)) ϵ and
for some constant C_5 > 0; and finally

    𝔞_ϵ^ 2𝒢 (ẑ)
      ⩽ C_6 exp (- C_7 | ẑ |) ,

when | ẑ | ⩾ 1 and for some constants C_6, C_7 > 0. It is
important to note that the constants C_4, C_5, C_6, C_7 are independent of
ϵ. The previous inequalities and eq. (<ref>) imply that
𝔼 [| ⟨η_ϵ, g ⟩ |^2] < + ∞ for any
ϵ⩾ 0 and | α | < 4 √(2)π (with η_0 =
η).

Let D_k, with k ⩾ - 1, be the functions related to Littlewood-Paley block (see Appendix <ref> for the definition
of this concept). Using Remark <ref>, we can suppose that there
exist some constants γ_- 1, γ > 0 and 0 < θ_- 1, θ
< 1 such that

    | D_- 1 (ẑ) | ≲exp (- γ_- 1 | ẑ |^θ_-
       1)   and   | D_k (ẑ) | ≲ 2^4 kexp (-
       γ 2^k θ | ẑ |^θ),

where k > 0. Using this decay at infinity, we get

    𝔼 [⟨η_ϵ, D_k (ẑ - ·) ⟩_L^2_r_ℓ^2] = ∫_ℝ^4𝔼 [⟨η_ϵ, D_k ⟩^2] (1 + | ẑ |)^- 2 ℓẑ≲𝔼 [⟨η_ϵ, D_k ⟩^2] < + ∞,

whenever α < 4 √(2)π, and where we used the invariance in law of
η_ϵ with respect to translations. Since D_k (x, z) = 2^4 k
D_0 (2^k x, 2^k z) and using (<ref>)
and (<ref>), we obtain

    𝔼 [⟨η_ϵ, D_k ⟩^2]     =    ∫_ℝ^8 D_k (ẑ) D_k (ẑ') exp (α^2 ·𝔞_ϵ^ 2𝒢 (ẑ -
      ẑ')) ẑẑ' 
       ≲   ∫_ℝ^8 | D_k (ẑ) D_k (ẑ') | | ẑ
      - ẑ' |^- 2 α^2/(4 π)^2ẑẑ'
       
           + ( 1 + ϵ^4 - 2 α^2/(4 π)^2)
      ∫_ℝ^8 | D_k (ẑ) D_k (ẑ') | ẑẑ'  
       ≲    2^2 α^2/(4 π)^2 k( ∫_ℝ^8( 1 + | ẑ - ẑ' |^- 2 α^2/(4 π)^2) |
      D_0 (ẑ) D_0 (ẑ') | ẑẑ' )
      ≲ 2^2 α^2/(4 π)^2 k

where all the constants implied in the symbol ≲ are independent of
ϵ. This means that

    𝔼 [η_ϵ^2_B^s_2, 2, ℓ] ≲∑^+
      ∞_k = - 1 2^2 α^2/(4 π)^2 k + 2 s k,

which is finite and uniformly bounded in ϵ whenever s < -
α^2/(4 π)^2 and for ℓ > 0 large enough. Furthermore we
have that

    𝔼 [η - η_ϵ_B^s_2, 2, ℓ^2]^1 / 2   ≲   ( ∑_k = - 1^+ ∞ 2^2 s k𝔼[| ⟨η - η_ϵ, D_k ⟩ |^2] )^1 / 2
       ≲   ∑_k = - 1^+ ∞ 2^s k∑^+ ∞_n = 0| α |^n/n!𝔼 [| ⟨ℐξ^♢ n -
      ℐξ_ϵ^♢ n, D_k ⟩ |^2]^1 / 2

for ℓ big enough. On the other hand we get

    ∑_k = - 1^+ ∞ 2^s k∑^+ ∞_n = 0| α
      |^n/n!𝔼 [| ⟨ℐξ^♢ n - ℐξ_ϵ^♢ n, D_k ⟩ |^2]^1 / 2   ⩽   ∑_k
      = - 1^+ ∞ 2^s k∑^+ ∞_n = 0| α |^n/n!𝔼 [| ⟨ℐξ^♢ n, D_k ⟩ |^2]^1 / 2
           + ∑_k = - 1^+ ∞ 2^s k∑^+ ∞_n = 0|
      α |^n/n!𝔼 [| ⟨ℐξ_ϵ^♢
      n, D_k ⟩ |^2]^1 / 2
       ⩽   1/2∑_k = - 1^+ ∞ 2^s k∑^+
      ∞_n = 0β^2 n | α |^2 n/(n!)^2𝔼 [|
      ⟨ℐξ^♢ n, D_k ⟩ |^2]  
           + 1/2∑_k = - 1^+ ∞ 2^s k∑^+ ∞_n =
      0β^2 n | α |^2 n/(n!)^2𝔼 [| ⟨ℐξ_ϵ^♢ n, D_k ⟩ |^2]  
           + β^2/β^2 - 1∑_k = - 1^+ ∞ 2^s k
       ⩽   1/2∑_k = - 1^+ ∞ 2^s k𝔼 [|
      ⟨ : exp (βαℐξ_ϵ^♢ n) :, D_k
      ⟩ |^2]  
           + 1/2∑_k = - 1^+ ∞ 2^s k𝔼 [| ⟨
      : exp (βαℐξ^♢ n) :, D_k ⟩ |^2] 
      
           + β^2/β^2 - 1∑_k = - 1^+ ∞ 2^s k < C

for some constant C > 0 independent of ϵ, whenever β > 1 is
arbitrary small and s < - βα^2/(4 π)^2, where we use an
estimate similar to (<ref>) and (<ref>) for the measure :
exp (βαℐξ^♢ n) :. Furthermore, since
𝔞_ϵ is a regular mollifier and by the properties of Wick
product, we obtain

    𝔼 [| ⟨ℐξ^♢ n - ℐξ_ϵ^♢ n, D_k ⟩ |^2] → 0,

as ε→ 0 and for any k ≥ - 1 and any n ∈ℕ. The above inequality, together with the Lebesgue dominated
convergence theorem, implies that

    𝔼 [η - η_ϵ_B^s_2, 2, ℓ^2]^1 / 2→ 0,

when ϵ→ 0, for any ℓ > 0 large enough. We have thus
proven the following lemma.


  For | α | < 4 √(2)π and s < -
  α^2/(4 π)^2 and ℓ > 0 large enough we have that
  η_ϵ→η as ϵ→ 0 in L^2
  (𝒲 ; B^s_2, 2, ℓ (ℝ^4), μ) and thus in
  probability in B^s_2, 2, ℓ (ℝ^4).


We want to use the previous lemma to prove the following theorem.


  For | α | < 4 √(2)π, 1 < p ⩽ 2, s
  < - α^2 (p - 1)/(4 π)^2 and ℓ > 0 large enough we have
  that η_ϵ→η, as ϵ→ 0, in
  L^p (𝒲 ; B^s_p, p, ℓ (ℝ^4), μ) and thus
  in probability in B^s_p, p, ℓ (ℝ^4).


In order to prove Theorem <ref> we introduce the following lemma.


  Let B_r (ẑ) be the ball of radius r and center
  in ẑ∈ℝ^4 then for any r < R, and for any 1 < p < 2
  
    𝔼[ ( ∫_B_r (ẑ)η_ϵ)^p ] ≲ r^- α^2/(4 π)^2 p (p - 1) + 4
         p

  where the constants depend only R and are uniform on ϵ→
  0 and | α | < 4 √(2)π.



  The proof can be found in Proposition 2.7 of <cit.>.


Proof of Theorem <ref>
  First of all we have that, for any ε > 0 and k ⩾ 0,     
    𝔼 [⟨η_ϵ, D_k (ẑ - ·) ⟩_L^p_ℓ^p] = ∫_ℝ^4𝔼 [| ⟨η_ϵ, D_k ⟩ |^p] (1 + | ẑ |)^- p ℓẑ≲𝔼 [| ⟨η_ϵ, D_k ⟩ |^p]
         .

  Then
  
    𝔼 [| ⟨η_ϵ, D_k ⟩ |^p] ≲𝔼[ ( ∫_B_r_k (0) | D_k (ẑ) | η_ϵ (ẑ) )^p ] +𝔼[ (
         ∫_ℝ^4 ∖ B_r_k (0) | D_k (ẑ) | η_ϵ (ẑ) )^2 ]^p / 2 .

  If we choose r_k = 2^- k (log (2^β k))^u, for some β, u >
  0, from Lemma <ref> we have, for any k ⩾ 0,   
    𝔼[ ( ∫_B_r_k (0) | D_k (ẑ) | η_ϵ (ẑ) )^p ] ≲ 2^4 k p
         2^α^2/(4 π)^2 p (p - 1) k - 4 p klog (2^β k)^u
         ( - α^2/(4 π)^2 p (p - 1) + 4 p )

  Furthermore, if we denote by γ > 0 and 0 < θ < 1 the real
  number such that D_0 (ẑ) ≲exp (- γ | ẑ
  |^θ) and choosing u = 1/θ' where 0 < θ' <
  θ, we obtain
  
    [                                                          𝔼[ ( ∫_ℝ^4 ∖ B_r_k (0) |
           D_k (ẑ) | η_ϵ (ẑ) )^2 ]^p /
           2≲;      ≲ 2^8 k - γβ k∫_| ẑ | > r_k, | ẑ |
           > r_k e^γβ k - γ 2^k θ (| ẑ |^θ +
           | ẑ' |^θ) + α^2 𝒢_ϵ (ẑ -
           ẑ')ẑẑ'; ≲ 2^8 k - γβ k∫_| ẑ | > r_k, | ẑ |
           > r_k e^- γ 2^k θ (| ẑ |^θ - θ' + |
           ẑ' |^θ - θ') + α^2 𝒢_ϵ
           (ẑ - ẑ')ẑẑ';                    ≲ 2^8 k - γβ k∫_ℝ^8 e^- γ (|
           ẑ |^θ - θ' + | ẑ' |^θ - θ') +
           α^2 𝒢_ϵ (ẑ - ẑ')ẑẑ' ≲ 2^8 k - γβ k ]

  If we choose β such that 8 - γβ < α^2/(4
  π)^2 p (p - 1) we obtain that
  
    𝔼 [| ⟨η_ϵ, D_k ⟩ |^p] ≲
        2^α^2/(4 π)^2 p (p - 1) klog (2^β k)^u ( -
        α^2/(4 π)^2 p (p - 1) + 4 p )

  uniformly in ϵ. This implies that 𝔼 [η_ϵ_B^s_p, p, ℓ^p] is bounded for s < - α^2/(4 π)^2
  (p - 1) uniformly in ϵ > 0.
  
  Furthermore, by Lemma <ref>, 𝔼 [| ⟨η_ϵ, D_k ⟩ |^2] is uniformly bounded. This means that the
  random variables | ⟨η_ϵ, D_k ⟩ |^p (for p < 2)
  are uniformly integrable. On the other hand, by Lemma <ref> and
  since D_k ∈ B^2_2, 2, ℓ, | ⟨η_ϵ, D_k ⟩
  |^p converges to | ⟨η, D_k ⟩ |^p in probability, we have
  that 𝔼 [| ⟨η_ϵ, D_k ⟩ |^p] →𝔼 [| ⟨η, D_k ⟩ |^p], as ε→
  0. Thus by the bound (<ref>) we can use the Lebesgue dominated
  converge theorem for computing the limit of 𝔼 [η_ϵ_B^s_p, p, ℓ^p], obtaining
  
    𝔼 [η_ϵ_B^s_p, p, ℓ^p] →𝔼 [η_B^s_p, p, ℓ^p],

  as ε→ 0. But, since 𝔼 [η_ϵ_B^s_p, p, ℓ^p] is uniformly bounded and since, by
  Lemma <ref>, as ε→ 0, η_ϵ→η in probability in B^- 2_2, 2, ℓ, we have that
  η_ϵ converges weakly to η in L^p (𝒲, B^s_p,
  p, ℓ, μ). Since L^p (𝒲, B^s_p, p, ℓ, μ) is a uniformly convex space, being the space of L^p functions taking
  values in the uniformly convex space B^s_p, p, ℓ
  (see <cit.>), by the weak convergence η_ϵ→η and the convergence of the L^p norm 𝔼 [η_ϵ_B^s_p, p, ℓ^p] →𝔼 [η_B^s_p, p, ℓ^p], we obtain that η_ϵ converges
  strongly to η in B^s_p, p, ℓ (ℝ^4), as ε→ 0. 




 §.§ Analysis of the elliptic SPDE


In this section we want to prove the following theorem.


  For any | α | < α_max (see equation (<ref>)), there are some p, s, δ∈ℝ such that 1 < p
  ⩽ 2 and p < 2 (4 π)^2/α^2, - 1 < s <
  α^2 (p - 1)/(4 π)^2 and 0 < δ < s + 1, for which
  equation (<ref>) admits a unique solution in B^s +
  2_p, p, ℓ for ℓ > 0 large enough and such that αϕ̅⩽ 0.  Furthermore we have that there exists a
  subsequence ε_n → 0 such that   ϕ̅_ϵ_n→ϕ̅ in B^s + 2 - δ_p,
  p, ℓ + δ' (for some δ' > 0 small enough), as n →∞, almost surely.


Hereafter, for t > 0, we write 𝔹^t_p, p, ℓ (ℝ^4) =
B^t_p, p, ℓ (ℝ^4) ∩ L^∞ (ℝ^4). The space
𝔹^t_p, p, ℓ has a natural norm given by the sum of the norms
of B^t_p, p, ℓ (ℝ^4) and L^∞ (ℝ^4).
Furthermore we can equip the space 𝔹^t_p, p, ℓ with a
different notion of convergence of sequences: we say that a sequence h_n ∈𝔹^t_p, p, ℓ converges to h ∈𝔹^t_p, p, ℓ if
h_n converges to h strongly in B^t_p, p, ℓ and sup_n  h_n
_∞ < + ∞ (it is important to note that h _∞⩽lim inf_n → + ∞ h_n _∞).


  For any t > 0 and 1 < p ⩽ 2, we have
  that 𝔹^t_p, p, ℓ⊂ B^t (p - 1)
  γ_q/γ, q/γ, ℓ (where 1 / p + 1 / q =
  1 and 0 < γ<1) and the immersion is continuous with respect to the
  natural convergence in 𝔹^t_p, p, ℓ. Furthermore
  𝔹^t_p, p, ℓ is a Banach algebra.



  By Proposition <ref>, the interpolation space
  (B^0_∞, ∞, B^t_p, p, ℓ)_p γ/q, q between
  L^∞⊂ B^0_∞, ∞ and B^t_p, p, ℓ is exactly
  B^t γp/q_q/γ, q/γ, ℓ = B^t
  γ (p - 1)_q/γ, q/γ, ℓ, where q =
  p - 1/p. This means that 𝔹^t_p, p, ℓ = L^∞∩ B^t_p, p, ℓ⊂ B^0_∞, ∞∩ B^s_p, p, ℓ is
  continuously embedded in B^t (p - 1) γ_q/γ,
  q/γ, ℓ. The fact that 𝔹_p, p, ℓ^t is an
  algebra is proven in Corollary 2.86 of <cit.>.



  Under the hypotheses of
  Theorem <ref> on α there are some p, s,
  δ such that 1 < p ⩽ 2 and p < 2 (4
  π)^2/α^2, - 1 < s < - ( α^2 (p - 1)/(4 π)^2∨ (p - 1) ) and 0 < δ < s + 1 and for which there is a 0 <
  γ < 1 such that
  
    (2 + s - δ) (p - 1)/p + s     >     0  
    s + 1/1 - δ <    γ    < 1  
    ( 1 - s + 1/4 p ) γ    <     p - 1.




  Here we introduce the new variables Z = 1 + s and k = p - 1. The
  statement of the lemma can be reformulated as follows: for any values of the
  quotient 0 < 1 - Z/k < 2 we can find 0 < Z < 1 and 0 < k < 1
  such that
  
    Z/1 - δ    <    k/( 1 - Z/4 (k + 1)
        )
    (1 + Z - δ) k/k + 1    >     1 - Z

  From the inequality (<ref>) we get that
  
    k > 1 - Z/2 Z - δ

  and inequality (<ref>) is equivalent to
  
    k > Z (4 - Z)/(Z^2 + 4 (1 - δ)) .

  This means that the thesis of the lemma holds if and only if k > G (Z,
  δ) where
  
    G (Z, δ) = max( Z (4 - Z)/(Z^2 + 4 (1 - δ)),
         1 - Z/2 Z - δ) = {[                 1 - Z/2 Z - δ  for Z ⩽ 4 - 2
           √(3); Z (4 - Z)/(Z^2 + 4 (1 - δ))  for   Z > 4 -
           2 √(3) ].

  for 0 < δ < 1 and δ/2 < Z < 1. The bounds on the
  possible α can be obtained as follows
  
    α_max^2/(4 π)^2 = sup_s, p- s/(p - 1) =
         sup_k, Z1 - Z/k = sup_δ/2 < Z < 1, 0 < δ
         < 11 - Z/G (Z, δ) = 1 - ( 4 - 2 √(3))/G ( 4 - 2 √(3), 0 ) = 8 - 4 √(3) .

  


We introduce the space 𝔐^s_p, p, ℓ⊂ B^s_p, p, ℓ
(with s, p and ℓ as in Theorem <ref>) which
is the set of Radon σ-finite measures μ contained in B^s_p, p,
ℓ such that | μ | ∈ B^s_p, p, ℓ, where | μ | = μ_+ +
μ_-, and where μ_+ and μ_- are the unique positive measures such
that μ = μ_+ - μ_-. We can define a notion of convergence on
𝔐^s_p, p, ℓ in the following way: a sequence μ_n ∈𝔐^s_p, p, ℓ converges to μ∈𝔐^s_p, p,
ℓ if μ_n converges strongly in B^s_p, p, ℓ and sup_n  |
μ_n | _B^s_p, p, ℓ < + ∞.

We now consider the natural product · between smooth functions which
are in 𝔹^s + 2 - δ_r, r, ℓ' and and smooth measures in
𝔐^s_p, p, ℓ.


  If s, p, δ, γ satisfy the thesis of
  Lemma <ref>, then the product · can be extended in a
  unique and associative way from 𝔹^s + 2 - δ_r, r, ℓ'×𝔐^s_p, p, ℓ into 𝔐^s_p, p, ℓ,
  where
  
    r = p/( ( 1 - s + 1/4 p ) γ + 1
         ) .

  This extension is (weakly) continuous with respect to the natural
  convergence in 𝔹^s + 2 - δ_r, r, ℓ' and
  𝔐^s_p, p, ℓ. Furthermore for any h ∈𝔹^s + 2
  - δ_r, r, ℓ' and μ∈𝔐^s_p, p, ℓ we have
  
    | h ·μ | _B^s_p, p, ℓ≲ h _∞· | μ | _B^s_p, p, ℓ .




  First of all we note that, by Proposition <ref>, the
  product · is well defined and (strongly) continuous as bilinear
  functional from B^(s + 2 - δ) (p - 1) r/p_p/p - 1,
  p/p - 1, ℓ'× B^s_p, p, ℓ into B^s_1, 1, ℓ +
  ℓ', where r = p/( ( 1 - s + 1/4 p )
  γ + 1 ) for some γ such that s + 1/1 - δ <
  γ < 1 and ( 1 - s + 1/4 p ) γ < p - 1 (whose
  existence is proven in Lemma <ref>).
  
  Indeed if γ satisfies the previous conditions, we have
  
    (s + 2 - δ) (p - 1) r/p + s > (s + 2 - δ) (p -
         1)/p + s > 0

  since by hypotheses r = p/( ( 1 - s + 1/4 p
  ) γ + 1 ) > 1 and (s + 2 - δ) (p - 1)/p + s
  > 0.
  
  The only thing that remains to prove is
  inequality (<ref>), since the other statements of the
  lemma easily follow from it. First of all, using the equivalent norm on
  Besov spaces of Proposition <ref>, we note that
  if μ is a Radon measure we have
  
    | μ | _B^s_p, p, ℓ∼( ∫_ℝ^4 ×
         (0, 1]( | μ | (B_λ (ẑ))/λ^s + n)^p r̅_ℓ (ẑ)^p ẑλ/λ)^1 / p .

  Using again Proposition <ref>, if h is a
  continuous bounded function we have that
  
    | h ·μ | _B^s_p, p, ℓ≲(
         ∫_ℝ^4 × (0, 1]( | h ·μ |
         (B_λ (ẑ))/λ^s + n)^p r̅_ℓ
         (ẑ)^p ẑλ/λ)^1 / p≲ h _∞· | μ | _B^s_p, p, ℓ .

  If h is a generic function on 𝔹^s + 2 - δ_r, r, ℓ'
  there exists a sequence of smooth functions such that, as n →∞, h_n → h in B^s + 2 - δ_r, r, ℓ' and h_n _∞⩽ h _∞. By the first part of the lemma
  we have that | h_n ·μ | converges weakly in 𝒮'
  (ℝ^4) to | h ·μ | from which we obtain
  
    | h ·μ | _B^s_p, p, ℓ⩽lim inf | h_n ·μ | _B^s_p, p, ℓ≲ | μ | _B^s_p, p, ℓlim inf h_n _∞≲ h _∞ | μ |
         _B^s_p, p, ℓ .



We introduce the following map

    𝒦_g (μ, φ̅) : = - α (- Δ_ẑ +
       m^2)^- 1 (g (z) G (αφ̅ (ẑ)) ·μ (ẑ)),

for any | α | <α_max, where G : ℝ→ℝ_+ is an increasing smooth bounded function with all
bounded derivatives such that G (x) = exp (x) for x ⩽ 0. It is
clear that if αϕ̅⩽ 0 and if ϕ̅ solves
equation (<ref>) then we have

    ϕ̅ = 𝒦_g (η, ϕ̅) .

Hereafter we denote by 𝔐^s_+, p, p, ℓ the space formed by
the positive distributions contained in 𝔐^s_p, p, ℓ.


  Let G be the function defined by φ⟼ G
  (φ), and let p, s, δ, γ be as in the thesis of Lemma
  <ref> then, when ℓ > 0 is big enough, we have that G is a
  continuous map from B^s + 2 - δ_p, p, ℓ into 𝔹^s +
  2 - δ_r, r, 2 ℓ where r is chosen as in Lemma
  <ref>.



  For any 0 ≤ k < 1, q > 1 and ℓ big enough we have that G is a
  continuous (actually Lipschitz) map from B^k_q, q, ℓ into
  𝔹^k_q, q, ℓ. We give a sketch of the proof for the weighted
  Besov spaces following the argument of <cit.> Section 5.5.
  
  This fact can be proven using the equivalent definition of Besov norm using
  finite differences (<ref>) for N = 0 and M = 1.
  Indeed we have that
  
    G (φ) _B^k_q, q, ℓ    =     G (φ) _L^p_ℓ
        + ( ∫_| h | < 1 | h |^- k q G (φ (x + h)) - G
        (φ (x)) _L^q_ℓ^q  h/| h |^n)^1/q
       ≲    ( G _∞ +  G' _∞) ( 1 + φ_L^p_ℓ + ( ∫_| h | < 1 | h |^- kΔ^1_h
        φ_L^q_ℓ^q  h/| h |^n)^1/q) 
       ≲    ( G _∞ +  G' _∞) (1 + φ_B^k_q, q, ℓ) .

  This proves that the map G takes values in 𝔹^k_q, q, ℓ. By
  using the real interpolation of Sobolev spaces
  
    (L^p_ℓ, W^1, p_ℓ)_s, q = B^s_p, q, ℓ,

  which is a consequence of the interpolation of unweighted Sobolev spaces
  (see <cit.> Theorem 6.2.4) and the isomorphism between weighted
  and unweighted Besov and Sobolev spaces (see <cit.> Theorem
  6.5), using the Lipschitz continuity of G in L^q_ℓ and in W^1,
  q_ℓ (see <cit.> for the proof in the unweighted
  Sobolev spaces), and exploiting a standard interpolation argument (see
  <cit.>), we obtain that G is also continuous.
  
  Consider φ∈ B^s + 2 - δ_p, p, ℓ, then ∇ G
  (φ) = G' (φ) ∇φ. By a Besov embedding theorem we
  have that φ∈ B^1 - ϵ_p', p', ℓ where p' = ( 1
  - s + 1/4 p ) for ε > δ small enough. By our
  arguments above this means that G' (φ) ∈𝔹^1 -
  ϵ_p', p', ℓ, and by Lemma <ref>, G'
  (φ) ∈ B^(1 - ϵ) γ_p'/γ,
  p'/γ, ℓ This implies that, for γ > s + 1/1 -
  ε and ( 1 - s + 1/4 p ) γ < p - 1 (we
  must choose δ < ε < s + 1/γ - 1 here), the
  product G' (φ) ∇φ is well defined and in B^s + 1 -
  δ_r, r, ℓ. By Remark <ref> below, this
  implies that G (φ) ∈𝔹^s + 2 - δ_r, r, ℓ.
  
  In a similar way, when φ_n ∈ B^s + 2 - δ_p, p, ℓ
  converges to φ∈ B^s + 2 - δ_p, p, ℓ, it is possible
  to prove that G (φ_n) → G (φ) in 𝔹^s + 2
  - δ_r, r, ℓ. Indeed, by Remark <ref>, we
  have
  
    G (φ) - G (φ_n) _B^s + 2 - δ_r, r, 2 ℓ   ≲    G (φ) - G (φ_n) _L^r_2 ℓ +  G'
        (φ) ∇φ - G' (φ_n) ∇φ_n _B^s + 1 -
        δ_r, r, 2 ℓ
       ≲    G (φ) - G (φ_n) _L^r_2 ℓ +  G'
        (φ) _B^(1 - ϵ) γ_p'/γ,
        p'/γ, ℓ∇φ - ∇φ_n _B^s +
        1 - δ_p, p, ℓ + 
           + sup_n (∇φ_n _B^s + 1 - δ_p, p, ℓ)
         G' (φ) - G' (φ_n) _B^(1 - ϵ)
        γ_p'/γ, p'/γ, ℓ→ 0

  since G'  is continuous from B^s + 2 - δ_p, p, ℓ into
  B^(1 - ϵ) γ_p'/γ, p'/γ, ℓ (as
  it was proven in the first part of the proof).



  For p, s, δ, γ satisfying the
  thesis of Lemma <ref>, for any ℓ > 0 big enough, for any
  fixed μ∈𝔐^s_+, p, p, ℓ and small enough δ' >
  0, there exists at least a solution φ̅∈ B^s + 2 -
  δ_p, p, ℓ + δ' to the equation
  
    φ̅ = 𝒦_g (μ, φ̅) .




  We want to use Schaefer's fixed-point theorem (see Theorem 4 Section 9.2
  Chapter 9 of <cit.>) to prove the lemma. In order to do this we
  have to prove that 𝒦_g is continuous in φ, that it maps
  any bounded set into a compact set and that the set of solutions to the
  equations
  
    φ̅ = λ𝒦_g (μ, φ̅)

  is bounded uniformly for all 0 ⩽λ⩽ 1.
  
  The continuity of 𝒦_g (in both μ and φ) is a
  consequence of Lemma <ref> and Lemma <ref>.
  Indeed in the cone 𝔐^s_+, p, p, ℓ the natural convergence
  in 𝔐^s_+, p, p, ℓ⊂𝔐^s_p, p, ℓ   coincides with the strong convergence in B^s_p, p, ℓ, since if μ∈𝔐^s_+, p, p, ℓ then | μ | = μ.
  
  Furthermore, by Lemma <ref>, the map φ→ G
  (φ) is continuous from B^s + 2 - δ_p, p, ℓ + δ'
  into 𝔹^s + 2 - δ_r, r, ℓ + δ' where r =
  p/( ( 1 - s + 1/4 p ) γ + 1 ).
  Thus we can apply Lemma <ref>, from which we obtain
  that the map φ⟼ G (φ) ·μ is continuous in
  φ. Finally the linear operator (- Δ_ẑ + m^2)^- 1 is
  continuous from B^s_p, p, ℓ into B^s + 2_p, p, ℓ and thus
  compact from B^s_p, p, ℓ into B^s + 2 - δ_p, p, ℓ +
  δ' (since, by Proposition <ref>, the immersion
  B^s + 2_p, p, ℓ↪ B^s + 2 - δ_p, p, ℓ +
  δ' is compact). Thus by Lemma <ref> and Lemma
  <ref>, the map 𝒦_g is weakly continuous as a map into
  B^s + 2_p, p, ℓ and strongly continuous as a map into B^s + 2 -
  δ_p, p, ℓ + δ'.
  
  The map 𝒦_g is compact since the following inequality holds
  
    𝒦_g (μ, φ) _B^s + 2_p, p, ℓ≲ G
        _∞μ_B^s_p, p, ℓ

  and, by Proposition <ref>, we have the compact
  immersion B^s + 2_p, p, ℓ↪ B^s + 2 - δ_p, p,
  ℓ + δ'.
  
  Finally the uniform boundedness in λ follows from
  inequality (<ref>). This proves the thesis of the
  lemma.



  Under the hypotheses of
  Theorem <ref> the solution to
  equation (<ref>) is unique in B^s + 2 - δ_p, p, ℓ +
  δ' for δ, δ' ⩾ 0 small enough.



  Let J : ℝ→ℝ be a smooth, bounded, strictly
  increasing function such that J (0) = 0 and J (- x) = - J (x), and let
  φ̅_1 and φ̅_2 be two solutions to
  equation (<ref>). By Lemma <ref> and Lemma
  <ref>, J (φ̅_1 - φ̅_2) ∈ B^(s + 2 -
  δ) p - 1/r p_q, q, ℓ + δ' which implies that
  r̅_ℓ' (λẑ) J (φ̅_1 - φ̅_2) ∈
  (B^s_p, p, ℓ)^ for δ' ⩾ 0 small enough,
  ℓ' > 0 large enough and any λ > 0. This means that
  
    ⟨r̅_ℓ' (λẑ) J (φ̅_1 -
         φ̅_2), (- Δ_ẑ + m^2) (φ̅_1 -
         φ̅_2 - 𝒦_g (μ, φ̅_1) + 𝒦_g
         (μ, φ̅_2)) ⟩ = 0.

  We are going to see that the inequality
  
    ⟨r̅_ℓ' (λẑ) J (φ̅_1 -
         φ̅_2), (- Δ_ẑ + m^2) (φ̅_1 -
         φ̅_2) ⟩⩾ C ∫r̅_ℓ' (λẑ) J (φ̅_1 - φ̅_2) (φ̅_1 -
         φ̅_2) ẑ

  holds for λ > 0 small enough and some constant C > 0. Indeed let
  f_1, f_2 be two smooth functions then, for λ > 0 small enough,
  
    [                                    ⟨r̅_ℓ' (λẑ) J (f_1 - f_2), (-
           Δ_ẑ + m^2) (f_1 - f_2) ⟩;                                   = ∫r̅_ℓ' (λẑ) J (f_1 - f_2) (-
           Δ_ẑ + m^2) (f_1 - f_2) ẑ;                                      = ∫r̅_ℓ' (λẑ) J' (f_1 - f_2) |
           ∇ f_1 - ∇ f_2 |^2 ẑ +;                                                + λ∫∇r̅_ℓ' (λẑ) J (f_1 - f_2) · (∇ f_1 - ∇ f_2) ẑ;                                             + m^2 ∫r̅_ℓ' (λẑ) J (f_1
           - f_2) (f_1 - f_2) ẑ; ⩾ - λ^2 ∫ (Δr̅_ℓ'
           (λẑ)) J^- 1 (f_1 - f_2) ẑ + m^2 ∫r̅_ℓ' (λẑ) J (f_1 - f_2) (f_1 - f_2) ẑ;                            ⩾∫( m^2 - | λ^2 Δr̅_ℓ'/r̅_ℓ'| ) ρ_ℓ' (λẑ) J (f_1 - f_2) (f_1 - f_2) ẑ;                                               ⩾ C ∫r̅_ℓ' (λẑ) J (f_1
           - f_2) (f_1 - f_2) ẑ ]

  where J^- 1 (t) = ∫_0^t J (τ) τ. For these deductions we
  used the fact that
  
    [ ∫∇r̅_ℓ' (λẑ) J (f_1 - f_2) ·
           (∇ f_1 - ∇ f_2) ẑ = ∫∇r̅_ℓ'
           (λẑ) ∇ J^- 1 (f_1 - f_2) ẑ =;                                                        = - λ∫Δr̅_ℓ' (λẑ) J^- 1 (f_1
           - f_2) ẑ ]

  which is true since J^- 1 is a Lipschitz function such that J^- 1 (0)
  = 0, and thus, by the first part of the proof of Lemma <ref>, J^-
  1 (f_1 - f_2) ∈ W^1, p_ℓ (ℝ^4). We have, also, exploited
  the fact that J^- 1 (t) ⩽ J (t) t since J is increasing and
  the fact that we can choose λ small enough such that the last
  inequality holds. This proves that
  
    ⟨r̅_ℓ' (λẑ) J (f_1 - f_2), (-
        Δ_ẑ + m^2) (f_1 - f_2) ⟩⩾ C ∫r̅_ℓ' (λẑ) J (f_1 - f_2) (f_1 - f_2) ẑ,

  for λ small enough, and some C > 0, for smooth functions f_1, f_2
  ∈ B^s + 2 - δ_p, p, ℓ + δ'. Since the expressions
  ⟨r̅_ℓ' (λẑ) J (f_1 - f_2), (-
  Δ_ẑ + m^2) (f_1 - f_2) ⟩ and ∫r̅_ℓ'
  (λẑ) J (f_1 - f_2) (f_1 - f_2) ẑ are continuous
  for f_1, f_2 ∈ B^2 + 2 - δ_p, p, ℓ + δ' (with respect
  to the B^s + 2 - δ_p, p, ℓ + δ-natural norm) we can
  extend inequality (<ref>) to the case of general
  functions f_1, f_2 ∈ B^s + 2 - δ_p, p, ℓ + δ' and so it
  holds in particular for f_1 = φ̅_1 and f_2 = φ̅_2.
  Furthermore we have that ⟨r̅_ℓ' (λẑ) J
  (φ̅_1 - φ̅_2), (- Δ_ẑ + m^2) (-
  𝒦_g (μ, φ̅_1) + 𝒦_g (μ, φ̅_2))
  ⟩⩾ 0. Indeed by Lemma <ref> the product
  is associative and thus we obtain
  
    ⟨r̅_ℓ' (λẑ) J (φ̅_1 -
        φ̅_2), (- Δ_ẑ + m^2) (- 𝒦_g (μ,
        φ̅_1) + 𝒦_g (μ, φ̅_2)) ⟩ =    
        = ∫r̅_ℓ' (λẑ) g (z) (α G (αφ̅_1) - α G (αφ̅_2)) J (φ̅_1 -
        φ̅_2) μ (ẑ) ⩾ 0,

  where we use that μ (ẑ) is a positive measure and (α
  G (α t_1) - α G (α t_2)) · J (t_1 - t_2) is positive
  since both G and J are increasing functions and J (0) = 0. Using the
  previous inequalities we obtain that
  
    ∫r̅_ℓ' (λẑ) J (φ̅_1 -
         φ̅_2) (φ̅_1 - φ̅_2) ẑ⩽ 0

  which holds only if φ̅_1 - φ̅_2 = 0, since J is a
  strictly increasing function. 



  Combining Lemma <ref> and
  Lemma <ref> we deduce that the map μ⟼φ̅, associating with the measure μ the unique
  solution φ̅ to equation (<ref>), is continuous with
  respect to μ. Indeed suppose that μ_n →μ in
  𝔐^s_+, p, p, ℓ (and so in B^s_p, p, ℓ), then by
  Lemma <ref>, we have that φ̅_n _B^s
  + 2_p, p, ℓ≲supμ_n _B^s_p, p, ℓ, and so
  there exists a converging subsequence φ̅_k_n, as n
  →∞. On the other hand, since 𝒦_g is continuous
  in both μ and φ, we get that φ̅_k_n converges to
  the unique solution φ to equation (<ref>) associated with
  μ . Since the limit does not depend on the subsequence we have that
  φ_n →φ strongly in B^s + 2 - δ_p, p, ℓ
  + δ', which proves the continuity of the solution map.
  
  This continuity of the solution map with respect to the measure μ is
  similar to the continuity result obtained for solutions of singular SPDEs
  defined by the methods of paracontrolled calculus or regularity structure
  theory. 


Proof of Theorem <ref>
  The existence and uniqueness of the solution to
  equation (<ref>) are proved in
  Lemma <ref> and
  Lemma <ref> considering the equation ϕ̅
  = 𝒦 (η, ϕ̅). What remains to prove is the convergence
  of ϕ̅_ε_n to ϕ̅, as ε_n
  → 0. Let ϵ_n → 0 be a sequence of positive
  numbers such that η_ϵ_n→η almost surely in
  𝔐^s_+, p, p, ℓ and let w ∈𝒲 be such that
  η_ϵ_n (w) →η (w) in 𝔐^s_+, p, p,
  ℓ, as ε→ 0. We note that
  
    ϕ̅_ϵ_n (w) = 𝔞^ 2_ϵ_n (𝒦_g (η_ϵ_n (w),
        ϕ̅_ε_n (w))) .

  From the equality (<ref>) and
  Lemma <ref> we obtain that
  
    ϕ̅_ϵ_n (w) _B^s + 2_p, p, ℓ≲sup_n ∈ℕη_ε_n_B^s_p, p, ℓ,

  uniformly in n. This means that that there exists a subsequence
  ϕ̅_ϵ_k_n (w) converging to some φ̅ in
  B^s + 2 - δ_p, p, ℓ + δ'. On the other hand we have that
  
    φ̅ = lim_n → + ∞ϕ̅_ϵ_k_n (w) = lim_n → + ∞𝔞^ 2_ϵ_n( 𝒦_g
         ( η_ϵ_k_n (w), ϕ̅_ε_k_n (w)
         ) ) = 𝒦_g (η (w), φ̅) .

  Since, by Lemma <ref>, equation (<ref>)
  has a unique solution we have that all the subsequences
  ϕ̅_ϵ_k_n (w) converge to the same φ̅ and
  so φ̅ = ϕ̅ (w) (which is the unique solution to
  equation (<ref>) evaluated in w ∈𝒲). Since
  the previous reasoning holds for almost every w ∈𝒲 and since
  η_ϵ_n→η almost surely, we have that
  ϕ̅_ϵ_n→ϕ̅ in B^s + 2 - δ_p,
  p, ℓ + δ' almost surely, as ε_n → 0.




 §.§ Dimensional reduction


In this section we want to prove the reduction principle for
equation (<ref>), i.e. that the random field ϕ (0, z) =
ℐξ (0, z) + ϕ̅ (0, z) has the law κ_g given by
expression (<ref>).

Before proving the dimensional reduction for
equation (<ref>) we have to prove that the restriction of a
solution to a two dimensional hyperplane is a well defined operation.


  There exists a version of the functional x ⟼ℐξ (x, ·) which is continuous as a function from
  ℝ^2 into 𝒞^0 -_ℓ (ℝ^2) ⊂ B^0
  -_p, p, ℓ + δ'. Furthermore for any h ∈ C^∞_0
  (ℝ^2) the sequence of random variables ∫_ℝ^2𝔞_ϵ (ℐξ) (z) h (z)  z
  converges to ⟨ℐξ (0, ·), h ⟩ in L^2 (μ), as ε→ 0.



  In order to prove that x ⟼ℐξ (x, ·) admits a
  continuous version we prove that for any smooth function h : ℝ^2
  →ℝ such that (- Δ_z + m^2)^- 1 / 2 + ϵ
  (h) ∈ L^2 (ℝ^2), with ε > 0, we have
  
    𝔼 [| ⟨ℐξ (x, ·) - ℐξ (y,
         ·), h ⟩ |^2] ≲ | x - y |^ϵ (- Δ_z +
         m^2)^- 1 / 2 + ϵ (h) _L^2 (ℝ^2),

  where the constants implied by ≲ do not depend on h, x, y. This
  result, exploiting hypercontractivity and a version of Kolmogorov continuity
  criterion for multidimensional random fields, implies the continuity of
  ℐξ (x, ·) with respect to x ∈ℝ^2. We note
  that, for any x, y ∈ℝ^2:
  
    𝔼 [| ⟨ℐξ (x, ·) - ℐξ (y,
         ·), h ⟩ |^2] = 2 ∫_ℝ^41 - e^i k_1 ·
         (x - y)/(k_1^2 + k_2^2 + m^2)^1 + 2 ϵĥ
         (k_2)^2/(k_1^2 + k_2^2 + m^2)^1 - 2 ϵ k_1  k_2 .

  Now we observe that | 1 - e^i k_1 · (x - y) | ⩽ 2 | x - y
  |^p - 1/p | k_1 |^p - 1/p for any p > 1. If we choose
  p = 1/1 - ϵ, 0 < ϵ < 1, we obtain the claim.
  
  In order to prove that ∫_ℝ^2𝔞_ϵ (ℐξ) (z) h (z)  z converges, as ε→ 0, to ⟨ℐξ (0, ·), h ⟩ in L^2
  (μ), we note that the distribution ℐ (δ_0 (x)
  · h (z)) belongs to L^2 (ℝ^4), implying that
  
    ∫_ℝ^2𝔞_ϵ (ℐξ)
         (z) h (z)  z = δ [ℐ (δ_0 (x) ·𝔞_ϵ h (z))]     ⟨ℐξ
         (0, ·), h ⟩ = δ [ℐ (δ_0 (x) · h (z))],

  where δ denote the Skorohod integral with respect to the white noise
  ξ. Since ℐ (δ_0 (x) ·𝔞_ϵ h (z)) converges to ℐ (δ_0 (x) · h (z)) in
  L^2 (ℝ^4) also this claim follows. 



  The operator T_0 : C^0_ℓ (ℝ^4) →
  C^0_ℓ (ℝ^2) given by T_0 (h) (·) = h (0, ·), for h
  ∈ C^0_ℓ (ℝ^4), can be uniquely extended in a continuous way
  as an operator from B^s + 2 - δ_p, p, ℓ + δ'
  (ℝ^4) into B^s + 2 - 2 / p - δ_p, p, ℓ + δ'
  (ℝ^2), when s, p, δ satisfies the thesis of
  Lemma <ref>, ℓ > 0 big enough and δ' > 0 small
  enough.



  If s, p, δ satisfies the thesis of Lemma <ref> then s +
  2 - 2/p - δ > 0. Under this condition the proof can be found
  in Section 4.4.1 and Section 4.4.2 of <cit.>, see also
  Section 18.1 of <cit.>.



  Let ϕ be the unique solution to
  equation (<ref>) then, if | α | < α_max (see equation (<ref>)), there exists ℓ > 0 such that for any measurable and
  bounded function F on B^0 -_p, p, ℓ + δ' (ℝ^2), for
  some δ' > 0 small enough, we have
  
    𝔼 [F (ϕ (0, ·))] = ∫_B^0 -_p, p, ℓ + δ'
        (ℝ^2) F (ω) κ_g (ω) .




  We prove the equality (<ref>) for the case in which F (ω) =
  F̃ (⟨ω, f_1 ⟩, …, ⟨ω, f_n
  ⟩) where F̃ is a bounded continuous function, and f_1,
  …, f_n ∈ C^∞_0 (ℝ^2). Since the functions of the
  previous form generate all the σ-algebra of Borel measurable
  functions on B^0 -_p, p, ℓ + δ', where δ' > 0, proving
  the theorem for functions of the previous form is equivalent to prove the
  theorem in general.
  
  Since equation (<ref>) holds, we have only to prove that
  𝔼 [F (ϕ_ϵ_n (0, ·))] →𝔼 [F
  (ϕ (0, ·))], as ϵ_n → 0 and where ϵ_n is
  any subsequence such that ϕ̅_ϵ_n→ϕ̅
  almost surely (whose existence is proved in
  Theorem <ref>) and ∫_B^0 -_p, p, ℓ +
  δ' (ℝ^2) F (ω) κ_ϵ (ω)
  →∫_B^0 -_p, p, ℓ + δ' (ℝ^2) F (ω)
  κ_g (ω), as ϵ_n → 0.
  
  The first convergence follows from the fact that ϕ_ϵ =
  ϕ̅_ϵ + 𝔞_ϵ (ℐξ). Indeed, by Theorem <ref>, as ϵ_n
  → 0,
  
    ⟨ϕ̅_ϵ_n (0, ·), f_i ⟩ = ⟨ϕ̅_ε_n, T^_0 (f_i) ⟩→⟨ϕ̅, T^_0 (f_i) ⟩ = ⟨ϕ̅
         (0, ·), f_i ⟩,

  for any i = 1, …, n, almost surely and where the continuity of T_0,
  proved in Lemma <ref>, is used. On the other hand, by
  Lemma <ref>, we have ⟨𝔞_ϵ (ℐξ) (0, ·), f_i ⟩→⟨ℐξ
  (0, ·), f_i ⟩ in probability, as ε_n → 0,
  and this implies that ⟨ϕ_ϵ_n (0, ·), f_i ⟩→⟨ϕ (0, ·), f_i ⟩ in probability, for any i
  = 1, …, n.
  
  Since the convergence in probability implies the one in distribution we get
  𝔼 [F (ϕ_ϵ_n (0, ·))] →𝔼 [F
  (ϕ (0, ·))]. Finally since κ_ϵ_n converges weakly to
  κ_g, as ϵ_n → 0, the thesis follows. 


Now we want to discuss what happens if we remove the cut-off g in
equation (<ref>) so we consider the equation

    (- Δ_ẑ + m^2) (ϕ) + αexp (αϕ - ∞) =
      ξ,

for the content of this equation see the discussion ate the beginning of the
present section. In order to distinguish between the solution to
equation (<ref>) and equation (<ref>)
we denote by ϕ_g the solution of the former and by ϕ the solution of
the latter. We use also the symbols ϕ̅_g = ϕ_g - ℐξ
and ϕ̅ = ϕ - ℐξ, with ℐ = (- Δ +
m^2)^- 1.


  For any |α|<α_max there exists a unique solution ϕ to
  equation (<ref>).  Furthermore, for any affine
  transformation Φ : ℝ^4 →ℝ^4 in the
  Euclidean group of ℝ^4, the random field Φ_ (ϕ)
  (ẑ) = ϕ (Φ (ẑ)) has the same law as ϕ (ẑ).



  The existence and uniqueness for the solution to
  equation (<ref>) can be proven as in
  Lemma <ref> and
  Lemma <ref>, since the estimates used to prove
  those lemmas do not depend on g.
  
  The invariance of the law of the solutions with respect to affine
  transformations in the Euclidean group of ℝ^4 follows from the
  invariance (in law) of the white noise ξ and of the left-hand-side of
  equation (<ref>) with respect to translations and
  rotations and from the uniqueness of the solution to
  equation (<ref>).



  Consider |α| <α_max (see equation (<ref>)), if g_n is an (increasing) sequence of
  cut-offs with compact support such that g_n ↑ 1 then
  ϕ̅_g_n→ϕ̅ in B^s + 2 - δ_p, p, ℓ
  + δ' (which means that ϕ_g_n→ϕ in B^0 -_p,
  p, ℓ + δ' (ℝ^4)). This means that the sequence of
  probability measures κ_g_n on B^0 -_p, p, ℓ + δ'
  (ℝ^2) converges weakly, as n →∞, to a
  probability measure κ which is invariant with respect to the natural
  action of the Euclidean group of ℝ^2 on B^0 -_p, p, ℓ +
  δ' (ℝ^2) and does not depend on the sequence of cut-offs
  g_n.



  The proof is essentially based on the fact that
  
    ϕ̅_g _B^s + 2_p, p, ℓ≲η_B^s_p,
         p, ℓ

  uniformly in g. From the previous inequality and some reasoning similar to
  the ones used in the proof of Theorem <ref> the
  convergence follows. The properties of the limit measure κ follow
  from the same properties of the solution ϕ to
  equation (<ref>) proved in
  Proposition <ref>.




§ ELLIPTIC QUANTIZATION OF THE P (Φ)_2
MODEL


In this section we discuss the elliptic stochastic quantization of the P
(φ)_2 model where P is a polynomial of even degree and satisfying
Hypothesis QC. In order to avoid technical details we consider only the case
P (φ) = φ^2 n/2 n for n ∈ℕ, the general
case being then a straightforward generalization. We consider the equation

    (- Δ_x - Δ_z + m^2) (ϕ) + f (x) ϕ^♢ 2 n - 1 = ξ

where z ∈ M =𝕋^2, ♢ stands for the Wick product and
ξ is a ℝ^2 ×𝕋^2 white noise.
Equation (<ref>) can be better understood if we consider the
equation for ϕ̅ : = ϕ - ℐξ, the usual Da
Prato-Debussche trick (introduced in <cit.>), obtaining
the equation that is

    (- Δ_x - Δ_z + m^2) (ϕ̅) + ∑_k = 0^2 n - 12
      n - 1k f (x) ·ℐξ^♢ k·ϕ̅^2 n - 1
      - k = 0.

Equation (<ref>) is expected to be well defined, since we expect
that the solution ϕ̅ is in H^1 (ℝ^2 ×𝕋^2) =
W^1, 2 (ℝ^2 ×𝕋^2), in L^2 n_f^1 / 2 n
(ℝ^2 ×𝕋^2) (where L^2 n_f^1 / 2 n is the
weighted L^2 n space with respect to the space weight f^1/2 n)
and in B^2 - δ_𝔭, 𝔭, ℓ (ℝ^2 ×𝕋^2), where 𝔭 =2 n/2 n - 1. Furthermore it is
well known that ℐξ^♢ k∈𝒞^-
δ_ℓ = B^- δ_∞, ∞, ℓ (ℝ^2 ×𝕋^2) for any ℓ, δ > 0 (see <cit.>). In general we
expect that equation (<ref>), and so equation (<ref>),
for any realization of the noise ξ admits multiple solutions. So we need a
notion of weak solution to equation (<ref>).

First of all we consider a fixed probability space 𝒲^e =
(𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚, where

    𝔚 = H^1 - δ_1 (ℝ^2 ×𝕋^2) ∩
       L^2 n - δ_2_f^1 / 2 n + δ_3 (ℝ^2 ×𝕋^2) ∩ B^2 - δ_4_𝔭, 𝔭
       (ℝ^2 ×𝕋^2),

and where δ, δ_1, δ_2, δ_3, δ_4 > 0 are small
enough (we give some more precise conditions in what follows, see inequalities
(<ref>), (<ref>), (<ref>),
(<ref>) and (<ref>)), that is the space where
(ℐξ, ℐξ^♢ 2, …, ℐξ^♢ 2 n, ϕ̅) ∈𝒲^e are defined. We consider a
distinguished subspace of 𝔚 namely

    𝔚 = H^1 (ℝ^2 ×𝕋^2) ∩ L^2
       n_f^1 / 2 n (ℝ^2 ×𝕋^2) ∩𝔚 .



   Hereafter, we will always use the following additional
  hypothesis on the cut-off f:
  
  
  Hypothesis Hf1. The function f satisfies Hypothesis Hf
  and furthermore f satisfies the following properties, for any x, y ∈ℝ^2 and α∈ℕ^2 with | α | ≤ 2
  
    0 < f (x)^± 1 < c f (x - y)^± 1exp (d | y |)

  
    | D^α f (x) | ⩽ e f (x),

  where c, e > 0 and d ⩾ 0 are some constants.
 
  Examples of functions f satisfying Hypothesis Hf1 can be found
  in <cit.>.
  
    
  The importance of Hypothesis Hf1 is that an equivalent norm for B^s_p,
  q, f^β (ℝ^2 ×𝕋^2), i.e. the weighted Besov
  space with weight (f (x))^β (see <cit.> for the
  precise definitions), when | s | < 2, - 1 ⩽β⩽ 1
  and p > 1, is given by
  
    u _B^s_p, q, f^β∼ f^β u _B^s_p, q

  (see <cit.> Theorem 4.4). This means that an analogous version of
  Proposition <ref> and Proposition
  <ref> holds also for the space B^s_p, q,
  f^β.  



  A probability measure ν^e on 𝒲^e is a
  weak solution to equation (<ref>) if the projection of ν^e
  on [𝒞^- δ_ℓ (ℝ^2 ×𝕋^2)]^2
  n, δ > 0, gives the law of (ℐξ, ℐξ^♢ 2, …, ℐξ^♢ 2 n - 1) (which is the
  law of a Gaussian noise with covariance (- Δ_x - Δ_z + m^2)^-
  2 and its Wick powers) and it is supported on the set of solutions to the
  equation
  
    (- Δ_x - Δ_z + m^2) (θ̅) + ∑_k = 0^2 n - 12 n - 1k f (x) ·σ_k ·θ̅^2 n - 1 - k =
        0,

  where x ∈ℝ^2, z ∈𝕋^2, (σ_1, …,
  σ_2 n, θ̅) ∈𝒲^e and σ_0 = 1, defined
  on 𝒲^e.
  
  The weak solution ν to equation (<ref>) associated with the
  weak solution ν^e to equation (<ref>), is the probability
  law on 𝒞^- δ_ℓ (ℝ^2 ×𝕋^2) +
  𝔚, δ > 0, given by the push-forward of ν^e with
  respect to the map σ_1 + θ defined on 𝒲^e.


We introduce a modified equation on 𝔚 given by

    (- Δ_x - Δ_z + m^2) (θ̅) + 𝔞^
      2_ϵ[ ∑_k = 0^2 n - 12 n - 1k f
      (x) ·σ_k ·θ̅^2 n - 1 - k] = 0

where 𝔞_ϵ is some regular enough mollifier on
𝕋^2 such that the operator 𝒜_ϵ =
𝔞_ϵ satisfies Hypothesis H𝒜 (for
example we can take 𝔞_ϵ as the Green function associated
with the operator (- ϵΔ_z + 1)^- k for k large enough),
ε > 0. Also in this case we consider a special subspace of
𝔚 given by

    𝔚_ϵ = 𝒜_ϵ (H^1
       (ℝ^2 ×𝕋^2)) ∩ L^2 n_f^1 / 2 n
       (ℝ^2 ×𝕋^2) ∩𝔚 .

It is important to note that 𝔚_ϵ⊂𝔚.


  Let θ̅∈𝔚_ϵ⊂𝔚 be a solution to
  equation (<ref>) then θ̅∈ B^2 -
  δ”_𝔭, 𝔭 (ℝ^2 ×𝕋^2)
  (with 0 < δ < δ” < δ_4 being δ_4 as in the
  definition of 𝒲^e) and
  
    θ̅_H^1^2 + θ̅_L^2 n_f^1 / 2 n^2
        n   ≲   [ ∑_k = 1^2 nσ_k _𝒞^-
        δ_ℓ]^β_1
    θ̅_B^2 - δ”_𝔭, 𝔭, f^-
        β≲𝒜_ϵ^- 2θ̅_B^2 -
        δ”_𝔭, 𝔭, f^- β   ≲   [
        θ̅_H^1 + θ̅_L^2 n_f^1 / 2 n +
        ∑_k = 1^2 nσ_k _𝒞^- δ_ℓ]^β_2

  for any 0 ⩽β < 1/2 n - (2 n - 1) δ_3, and where
  β_1, β_2 ∈ℝ_+ and where β_1, β_2 and the
  constants implied by the symbol ≲ do not depend on ϵ,
  σ_k and β.



  Let 0 < γ < 1 be such that
  
    (2 n - 1) ( γ/2 n + 1 - γ/2) < 1.

  Then by Remark <ref>, Proposition <ref> and
  Proposition <ref> it is simple to see that
  θ̅∈ B^(1 - γ)_p, 2, f^1 / 2 n (ℝ^2 ×𝕋^2), where p = ( γ/2 n + 1 - γ/2)^- 1, and
  
    θ̅_B^(1 - γ)_p, 2, f^1 / 2 n≲θ̅_L^2 n_f^1 / 2 n^γ·θ̅^1 - γ_H^1 .

  Using the fact that θ̅∈𝒜_ϵ (H^1) we can
  multiply both sides of equation (<ref>) by
  𝒜_ϵ^- 2 (θ̅) and take the integral over
  ℝ^2 ×𝕋^2 obtaining
  
    𝒜_ϵ^- 1 (θ̅) _H^1^2    ⩽    -
        θ̅_L^2 n_f^1 / 2 n^2 n - ∑_k = 1^2 n - 12 n - 1k∫ f (x) ·σ_k (x, z) ·θ̅^2
        n - k (x, z)  x  z 
       ⩽    - θ̅_L^2 n_f^1 / 2 n^2 n + C
        ∑_k = 1^2 n - 1 f^1 / 2 nσ_k _𝒞^- δ[ θ̅_L^2 n_f^1 / 2 n^γθ̅^1 - γ_H^1]^2 n - k

  where we choose δ < 1 - γ and for some constant C > 0
  depending only on n. Using the condition (<ref>), the fact
  that θ̅_H^1≲𝒜^- 1_ε
  (θ̅) _H^1 and Young's inequality for products we obtain
  
    θ̅_H^1^2 + θ̅_L^2 n_f^1
        / 2 n^2 n≲∑_k = 1^2 n - 1 f^1 / 2 nσ_k
        _𝒞^- δ^1 / ϵ .

  for ϵ = ( 1 - (2 n - 1) ( γ/2 n + 1 -
  γ/2) ). Let
  
    0 < β < 1/2 n  and  δ < δ” < (1 -
        γ),

  we have
  
    [                                                     (𝒜_ϵ^- 2θ̅) f^- β_B^2 -
          δ”_𝔭, 𝔭                                                                                         ≲ (f (x))^-
          β[ ∑_k = 0^2 n - 12 n - 1k f (x) ·σ_k ·θ̅^2 n - 1 - k] _B^-
          δ”_𝔭, 𝔭;                                                                                                                                                                                   ≲                       ∑_k = 0^2 n - 1 (f (x))^1 - β·σ_k ·θ̅^2 n - 1 - k_B^-
          δ”_𝔭, 𝔭;                                                                                                                                                                                   ≲                                                 ∑_k = 0^2 n - 1 f^1 / 2 n - β·σ_k _𝒞^- δ×;                                                                                                                                                                                                                             ×[ θ̅_L^2 n_f^1 / 2 n^γ·θ̅^1 - γ_H^1]^2 n - 1 - k;                                                                                                                                                                                   ≲        ( θ̅_H^1 + θ̅_L^2
          n_f^1 / 2 n + ∑_k = 1^2 n - 1 f^1 / 2 n - βσ_k _𝒞^- δ)^β_2 ]

  where δ > 0 such that δ > δ”. Inserting now inequality
  (<ref>) into inequality (<ref>), and using the fact that θ̅_B^2 - δ”_𝔭, 𝔭, f^- β≲ (𝒜_ϵ^- 2θ̅) f^- β_B^2
  - δ”_𝔭, 𝔭 we obtain the thesis.


An easy consequence of the previous lemma is the following one.


  Let ℱ_ϵ : (𝒞^-
  δ_ℓ)^2 n - 1→𝒫
  (𝔚̅_ε), ε > 0, be the set-valued
  map associating (σ_1, …, σ_2 n - 1) with the set of
  solutions to equation (<ref>) in
  𝔚_ε. If K ⊂ (𝒞^-
  δ_ℓ)^2 n - 1 is a compact set then
  ⋃_ϵ < 1ℱ_ϵ (K) is a subset
  of 𝔚̅ and it is compact in 𝔚.



  The proof of the lemma consists only in noticing that a consequence of
  Proposition <ref> is that the inclusion map i : B^2
  - δ”_𝔭, 𝔭, f^- β∩ L^2 n_f^1 / 2
  n→𝔚 is compact, when β satisfies condition
  (<ref>) and it is large enough and δ” > δ > 0 is
  small enough. Indeed using Proposition <ref> and
  Proposition <ref>, it is simple to prove that
  B^2 - δ”_𝔭, 𝔭, f^- β is compactly
  imbedded in B^2 - δ_4_𝔭, 𝔭 when
  
    δ < δ” < δ_4

  since f^- β→ + ∞ as x → + ∞.
  
  Recalling that, by Remark <ref> and Proposition
  <ref>, L^p_f^1 / 2 n⊂ B^0_p, p, f^1 / 2
  n when p ⩾ 2, we obtain, using Proposition
  <ref>, that B^2 - δ”_𝔭,
  𝔭, f^- β∩ L^2 n_f^1 / 2 n is compactly embedded
  in B^(2 - δ”) θ - ϵ_p_θ - ϵ, p_θ
  - ϵ, f^β_θ + ϵ where
  
    p_θ    =    2 n/(2 n - 2) θ + 1
    β_θ    =     - θ( 1/2 n + β) +
        1/2 n

  for any 0 ⩽θ⩽ 1. Taking 0 < θ <
  δ”/(2 n - δ”) (2 n - 2), β = 0 and ϵ
  small enough we have that B^2 - δ”_𝔭, 𝔭,
  f^- β∩ L^2 n_f^1 / 2 n is compactly embedded in L^2 n -
  δ_2_f^1 / 2 n + δ_3 for any δ_2 < 1 and δ_3 >
  0.
  
  Furthermore if we take
  
    1 - δ_1/2 - δ” < θ < n - 1/2 n - 2

  
    ( 2 - δ”/1 - δ_1) 1/2 n -
         1/2 n < β < 1/2 n,

  which are a non-void conditions whenever
  
    δ < δ” < 1

  
    0 < δ < δ” < δ_1 2 n - 2/n - 1,

  we obtain that B^(2 - δ”) θ - ϵ_p_θ - ϵ,
  p_θ - ϵ, f^β_θ + ϵ compactly embeds in
  H^1 - δ_1 for ϵ small enough.
  
    
  Using inequalities (<ref>) and fact that the balls of (B^2 -
  δ”_𝔭, 𝔭, f^- β∩ L^2 n_f^1 / 2
  n) are closed sets of 𝔚, since the embedding map i : B^2 -
  δ”_𝔭, 𝔭, f^- β∩ L^2 n_f^1 / 2
  n→𝔚 is injective and continuous,
  ⋃_ϵ < 1ℱ_ϵ (K) is contained
  in 𝔚.


Hereafter we denote by 𝔓 :𝒲^e →
(𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n the
natural projection of the Cartesian product 𝒲^e = (𝒞^-
δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚
on the first component.


  Let ν_ϵ^e be a sequence of weak
  solutions to equation (<ref>) such that ν^e_ϵ
  ((𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚̅_ϵ) = 1, and suppose that
  𝔓_ (ν_ϵ^e) is tight. Then
  ν_ϵ^e is tight. Furthermore suppose that
  𝔓_ (ν_ϵ^e) weakly converges to the law of
  (ℐξ, ℐξ^♢ 2, …, ℐξ^♢ 2 n - 1) as ϵ→ 0, then any convergent
  subsequence ν_ϵ_n^e converges, as ε_n →
  0, to a weak solution ν^e to equation (<ref>) such that
  ν^e ((𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚) = 1.



  The proof of the tightness of ν_ϵ^e is similar to the one of
  Lemma <ref>, where Lemma <ref> is replaced by
  Lemma <ref>. The hypothesis that ν^e_ϵ
  ((𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚̅_ϵ) = 1 enters in the proof in the
  following way: let ℱ_ϵ : (𝒞^-
  δ_ℓ)^2 n - 1→𝒫 (𝔚),
  ε > 0, be the set-valued map associating (σ_1, …,
  σ_2 n - 1) with the set of solutions to equation (<ref>)
  in 𝔚, let K̃⊂ (𝒞^- δ_ℓ)^2
  n be a compact such that μ_ε (K̃) > 1 -
  ϵ̃, where μ_ε are the probability
  distributions of (ℐξ_ε, ℐξ_ε^♢ 2, …, ℐξ_ε^♢ 2 n) which are tight since μ_ε
  converges to μ the probability distribution of (ℐξ,
  ℐξ^♢ 2, …, ℐξ^♢ 2 n), as
  ε→ 0. Furthermore denote by 𝔎∪_ϵℱ_ϵ (K̃), which is
  compact by Lemma <ref>, then
  
    ν_ε^e (K̃×𝔎) ⩾ν_ε^e (K̃×∪_ϵℱ_ϵ (K̃)) ⩾ν_ε^e
         (K̃×ℱ_ϵ (K̃)) =
         ν_ε^e (K̃×ℱ_ϵ
         (K̃)) = μ_ε (K̃) ⩾ 1 -
         ϵ̃,

  where in the fourth passage we used that ν_ε (K̃×ℱ_ϵ (K̃)) = ν_ε (K̃×ℱ_ϵ (K̃)) since
  ν^e_ϵ ((𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚̅_ϵ) = 1.
  
  Since 𝔓_ (ν_ϵ^e) weakly converges, as
  ε→ 0, to the law of (ℐξ, ℐξ^♢ 2, …, ℐξ^♢ 2 n - 1) and so
  𝔓_ (ν^e) has the same law as (ℐξ,
  ℐξ^♢ 2, …, ℐξ^♢ 2 n - 1),
  the proof of the fact that ν^e is a weak solution to
  equation (<ref>) is equivalent to prove that ν^e is such
  that for any C^1 bounded function F  from B^-
  δ_4_𝔭, 𝔭 into ℝ with bounded
  derivative we have
  
    ∫ F ( (- Δ_x - Δ_z + m^2) (θ̅) + 
        ∑_k = 0^2 n - 12 n - 1k f (x) σ_k θ̅^2 n
        - 1 - k) ν^e = ∫ F (E) ν^e = F (0) .

  We know that ∫ F (E_ϵ) ν_ϵ^e = F (0) with
  
    E_ϵ = (- Δ_x - Δ_z + m^2) (θ̅) +
         𝔞^ 2_ϵ( ∑_k = 0^2 n
         - 12 n - 1k f (x) ·σ_k ·θ̅^2 n - 1
         - k) .

  On the other hand E_ϵ converges to E, as ε→ 0, uniformly on compact sets (since
  𝔞_ϵ^ 2 strongly converges as an
  operator to the identity on B^- δ_4_𝔭, 𝔭),
  which implies that F ∘ E_ϵ converges to F ∘ E uniformly
  on compact sets, as ε→ 0.
  
    
  Finally since the sets of the form ⋃_ϵ < 1ℱ_ϵ (K) are subsets of 𝔚̅, they are
  closed and they have an arbitrary measure we have that any limit ν^e is
  such that ν^e ((𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚) = 1.


This fact and the boundedness of F and of its derivatives implies (using a
reasoning similar to the one of Lemma <ref>, see also Lemma 2
in <cit.>) that F ∘ E_ϵν_ϵ^e weakly converges to F ∘ E ν^e, as
ε→ 0. This implies equation (<ref>) and thus
the thesis of the lemma.


  There exists at least one weak solution
  ν^e to equation (<ref>) such that, for any continuous
  bounded function F : B^- δ_𝔭, 𝔭, ℓ
  (𝕋^2) →ℝ (where δ > 0 and ℓ > 0 as
  in the definition of 𝒲^e and  𝔭 = 2 n/2 n -
  1),
  
    ∫_𝒲^e F (ϕ (0, ·)) Υ̃_f (σ,
        θ̅) ν^e (σ, θ̅) = Z_f ∫_B^-
        δ_𝔭, 𝔭, ℓ (𝕋^2) F (ω)
        κ (ω)

  where ϕ = σ_1 + θ̅ (where σ_1 and θ̅
  are as in Definition <ref>),
  
    Υ̃_f (σ, θ̅) = exp( ∑_k = 0^2
         n2 nk⟨σ_k θ̅^2 m - k, f' ⟩),

  
    Z_f = ∫_𝒲^eΥ̃_f (σ, θ̅)
         ν^e (σ, θ̅)

  and κ is given by
  
    κ/μ^- Δ_z = exp(
         ∫_𝕋^2ω^♢ 2 m (z)/2 m z
         )/Z_κ,

  where μ^- Δ_z is the Gaussian with covariance given
  by (<ref>) with 𝔏 = - Δ_z and
  𝒜 = id.



  Let ν^e_ϵ be a sequence of weak solutions
  to equation (<ref>) such that 𝔓_
  (ν_ϵ^e) ∼ (𝔞_ϵℐξ, …, (𝔞_ϵℐξ)^♢ 2 n - 1) (where ∼ means “with the same
  law”) and ν^e_ϵ ((𝒞^- δ_ℓ (ℝ^2
  ×𝕋^2))^2 n×𝔚̅_ϵ) = 1 then
  
    ∫exp( p ∑_k = 0^2 n2 nk⟨σ_k,
         f'  θ̅^2 n - k⟩) ν_ϵ^e < C_p

  form some constant C_p uniform in ϵ small enough and depending on
  p ≥ 1.


Proof
  We want to use Nelson's trick to prove the theorem (see Chapter V
  of <cit.>). Then we put E (σ_k, θ̅) = ∑_k =
  0^2 n2 nk⟨σ_k, f'  θ̅^2 n - k⟩ and introduce the following expression
  
    E_ϵ, N (σ_1, θ̅) = .
         {[ ∫_ℝ^2 ×𝕋^2∑_k = 0^2 n2
           nk( 𝔞_1/N - ϵ (σ_1) (ẑ)
           )^♢ k f' (x)  θ̅^2 n - k (ẑ) ẑ,                                                                                                                                                                              for ϵ < 1/N,;              ∫_ℝ^2 ×𝕋^2∑_k = 0^2 n2
           nk (σ_1 (ẑ))^♢ k f' (x)  θ̅^2 n - k
           (ẑ) ẑ,                                                                                                                                                                                for ϵ⩾1/N. ]. .

  We have to prove that, for any p ≥ 1,
  
    ( ∫ | E - E_ϵ, N |^p ν_ϵ^e )^1 /
        p≲ (p - 1)^u N^- α

  for some u ∈ℕ and α > 0 and
  
    E_ϵ, N≲ - f' (x) (log (N))^α',

  for some α' > 0, and that both inequalities are uniform in
  ϵ. Inequality (<ref>) is obvious when ϵ⩾1/N, since then E - E_ϵ, N = 0 ν^e_ϵ-almost
  surely. Consider the case ϵ < 1/N then
  
    | E - E_ϵ, N |    ≲   | ∑_k = 0^2 n∫_ℝ^2 ×𝕋^2 f' (x) ( σ_k (z) -
        ( 𝔞_1/N - ϵ (σ_1) (ẑ)
        )^♢ k) ·θ̅^2 n - k (ẑ) ẑ| 
       ≲   ∑_k = 1^2 n | f' (x) |^1 / 2 n(
        σ_k (z) - ( 𝔞_1/N - ϵ (σ_1)
        (ẑ) )^♢ k) _𝒞^- δ· | f' (x) |^1 - 1 / 2 nθ̅_B^δ_𝔭, 𝔭^2 n - k

  where 𝔭 = 2 n - 1/2 n. On the other hand, by
  Lemma <ref>, we have:
  
    | f' (x) |^1 - 1 / 2 nθ̅_B^δ_𝔭,
         𝔭^2 n - k≲( θ̅_L^2
         n_f^1 / 2 n + θ̅_H^1)^2 n - k≲( ∑_k = 1^2 nσ_k _𝒞^- δ_ℓ)^β_1 (2 n - 1),

  ν^e_ϵ-almost surely and uniformly in ϵ (since
  ν^e_ϵ ((𝒞^- δ_ℓ (ℝ^2 ×𝕋^2))^2 n×𝔚̅_ϵ) = 1). Using
  Hölder's inequality we obtain, for any p ≥ 1,
  
    ( ∫ | E - E_ϵ, N |^p ν_ϵ^e
        )^1/p   ≲   ∑_k = 1^2 n[ ∫
        | f' (x) |^1 / 2 n[ σ_k (z) - (
        𝔞_1/N - ϵ (σ_1) (ẑ)
        )^♢ k] _𝒞^- δ^2 pν^e_ϵ]^1/2 p×
          ×( ∫( ∑_k = 1^2 nσ_k
        _𝒞^- δ_ℓ)^2 β_1 (2 n - 1) pν_ϵ^e )^1/2 p

  Since the law of σ_k is a multilinear functional of a Gaussian random
  field, by hypercontractivity we have that there exists m_1, m_2 ∈ℕ and α > 0 such that
  
    ∫ | f' (x) |^1 / 2 n( σ_k (z) - (
        𝔞_1/N - ϵ (σ_1) (ẑ)
        )^♢ k) _𝒞^- δ^2 pν^e_ϵ   ≲    (p - 1)^p u_1 N^- α
    ∫( ∑_k = 1^2 nσ_k _𝒞^-
        δ_ℓ)^2 β_1 (2 N - 1) pν_ϵ^e    ≲    (p - 1)^p u_2

  uniformly in ϵ. This proves inequality (<ref>). We note
  that
  
    ∑_k = 0^2 n2 nk( 𝔞_1/N -
         ϵ (σ_1) (ẑ) )^♢ k f' (x) 
         θ̅^2 n - k (ẑ) = f' (x) H_2 n(
         𝔞_1/N - ϵ (σ_1) (ẑ) + θ̅
         (ẑ) ; c_N, ϵ)

  where H_2 n (x ; λ) is the Hermite polynomial of degree 2 n of a
  Gaussian random variable with variance λ and c_N, ϵ
  =𝔼[ ( 𝔞_1/N (ℐξ) )^2 ] ∼log (N), as N → + ∞ for any
  ϵ > 0. It is simple to see that, for any x ∈ℝ^2 and
  ẑ = (x, z) ∈ℝ^2 ×𝕋^2,
  
    f' (x) H_2 n( 𝔞_1/N - ϵ (σ_1)
        (ẑ) + θ̅ (ẑ) ; c_N, ϵ)    ≲   
        a_n f' (x) ( 𝔞_1/N - ϵ (σ_1)
        (ẑ) + θ̅ (ẑ) )^2 n
           + f' (x) ∑_k = 1^m a_k ( 𝔞_1/N -
        ϵ (σ_1) (ẑ) + θ̅ (ẑ) )^2 N - 2
        k c_N, ϵ^k 
       ≲   ã_n f' (x) ( 𝔞_1/N -
        ϵ (σ_1) (ẑ) + θ̅ (ẑ) )^2 N - f'
        (x) c_N, ϵ^n 
       ≲    - f' (x) c_N, ϵ^n ≲ - f' (x) (log (N))^n,

  where we used that f' ⩽ 0 and a_N, ã_N' > 0. This proves
  inequality (<ref>).
  
  Now we use Lemma V.5 of <cit.> and the fact that
  inequality (<ref>) and (<ref>) are uniform in ϵ,
  to prove that there exist two constants b, α” > 0 depending only on
  f' and m but not on ϵ > 0 such that
  
    ν_ϵ^e (E (σ, θ̅) > b log (K)) ⩽ e^-
        K^α” .

  Inequality (<ref>) implies that exp (E) ∈ L^p
  (ν^e_ϵ) for any p ≥ 1, and we have also a uniform bound on
  the L^p norm of exp (E) with respect to ϵ > 0. This concludes
  the proof of the lemma.


Proof of Theorem <ref>
  First we introduce the equation
  
    (- Δ_x - Δ_z + m^2) (ϕ) +
        𝔞_ε^ 2 (H_2 n (ϕ ;
        c_ϵ)) = 𝔞_ϵξ,

  where c_ϵ =𝔼 [(𝔞_ϵξ)^2]. Equation (<ref>) is of the
  form (<ref>), and the potential V_ϵ (y) = H_2 n (y ;
  c_ϵ), where y ∈ℝ, satisfies Hypothesis QC since it
  is an even polynomial with positive leading coefficient. This means that,
  for any ϵ > 0, there exists a weak solution ν_ϵ to
  equation (<ref>), such that
  
    ∫_𝒲 F (ϕ (0, ·)) Υ_f^ϵ (ϕ)
         ν_ϵ = ∫_B^- δ_𝔭, 𝔭,
         ℓ (𝕋^2) F (ω) κ_ϵ (ω)

  where Υ_f^ϵ (ϕ) = ∫_ℝ^2 ×𝕋^2 f' (x) H_2 n (ϕ (ẑ) ; c_ϵ) ẑ. On the other hand if by ν^e_ϵ we denote the law of
  (𝔞_ϵℐξ, …,
  (𝔞_ϵℐξ)^♢ 2
  n, ϕ - 𝔞_ϵℐξ)
  (where ϕ has law ν and 𝔞_ϵℐξ is given by equation (<ref>)), we have
  that ν^e_ϵ is a solution to equation (<ref>) and
  
    ∫_𝒲 F (ϕ (0, ·)) Υ_f^ϵ (ϕ)
         ν_ϵ = ∫_𝒲^e F (ϕ (0, ·))
         Υ̃_f (σ, θ̅) ν^e_ϵ
         (σ, θ̅) .

  Using the Galerkin approximation of Section <ref> and a
  method similar to the one of Lemma <ref>, it is possible to
  prove that ν^e_ϵ ((𝒞^- δ_ℓ (ℝ^2
  ×𝕋^2))^2 n×𝔚_ε) =
  1.
  
  By Lemma <ref> and Lemma <ref> we have that the measure
  F (ϕ (0, ·)) ν_ϵ^e weakly converges to F (ϕ
  (0, ·)) ν^e, as ε→ 0. Let G_r :
  𝒲^e →R, r ∈ℕ, be a sequence of
  continuous functions such that 0 ⩽ G_r ⩽ 1, G_r = 1 when
  Υ̃_f (σ, θ̅) ⩽ r and G_r = 0 when
  Υ̃_f (σ, θ̅) ⩾ 2 r (the existence of
  this kind of functions follows from the fact that Υ̃_f
  (σ, θ̅) is continuous on 𝒲^e). Since F (ϕ
  (0, ·)) ν_ϵ^e weakly converges to F (ϕ (0,
  ·)) ν^e then ∫_𝒲^e G_r F Υ̃_f
  ν^e_ϵ→∫_𝒲^e G_r F
  Υ̃_f ν^e, as ε→ 0. On the
  other hand, by Lemma <ref>, for all ε > 0 and any
  r > 0,
  
    | ∫_𝒲^e (G_r - 1) F Υ̃_f ν^e_ϵ| ⩽∫_Υ̃_f ⩾ r
         | F | Υ̃_f ν^e_ϵ⩽C_p  F
         _∞/r^p - 1

  for any p ≥ 1 (and a similar inequality is true for ϵ = 0).
  Taking r → + ∞ the thesis follows.




§ BESOV SPACES


In this appendix we recall some results on weighted Besov spaces used in this
paper. We consider only the case of Besov spaces defined on ℝ^n but
all what follows holds also for Besov spaces on 𝕋^n or on
ℝ^n_1×𝕋^n_2 (like the ones used in Section
<ref>).


First we recall the definition of Littlewood-Paley block: let χ, φ
be smooth non-negative functions from ℝ^n into ℝ such
that

  
  * supp (χ) ⊂ B_4/3 (0) and supp
  (φ) ⊂ B_8/3 (0) ∖ B_3/4 (0),
  
  
  * χ, φ≤ 1 and χ (y) + ∑_j ≥ 0φ (2^-
  j y) = 1 for any y ∈ℝ^n,
  
  
  * supp (χ) ∩supp (φ (2^- i·)) =
  ∅ for i ⩾ 1,
  
  
  * supp (φ (2^- j·)) ∩supp (φ (2^-
  i·)) = ∅ if | i - j | > 1,

where by B_r (x) we denote the ball of center x ∈ℝ^n and
radius r > 0.

We introduce the following notations: φ_- 1 = χ, φ_j
(·) = φ (2^- j·), D_j = φ̂_j and for any f ∈𝒮' (ℝ^n) we put Δ_j (f) = D_j  f.
Furthermore we write, for any ℓ > 0,  r_ℓ (y) = (1 + | y |^2)^-
ℓ / 2, L^p_ℓ (ℝ^d) is the L^p space with respect to the
norm

    f _p, ℓ = ( ∫_ℝ^n (f (y) r_ℓ (y))^p
        y )^1 / p,

where p ∈ [1, + ∞].


  Consider s ∈ℝ, p, q ∈ [1, + ∞]
  and ℓ∈ℝ. If f ∈𝒮 (ℝ^n) we define
  the norm
  
    f _B^s_p, q, ℓ = ( ∑_j ≥ - 1 2^s qΔ_j
        (f) _p, ℓ^q )^1 / q .

  The space B^s_p, q, ℓ is the subset of 𝒮' (ℝ^n)
  such that the norm (<ref>) is finite. 



  It is important to note that B^s_2, 2, ℓ is equal to the weighted
  Sobolev space W^s, 2_ℓ = H^s_ℓ of L^2 distributions, and
  that for s > 0, s ∉ℕ, B^s_∞, ∞ is equal
  to 𝒞^s, the space of Hölder functions of regularity s.
  Moreover we remark that B^s_p, q = B^s_p, q, 0.



  It is always possible to choose χ and φ in
  such a way that there exists some constants γ_- 1, γ > 0 and
  0 < θ_- 1, θ < 1 such that
  
    | D_- 1 (y) | ≲exp (- γ_- 1 | y |^θ_- 1)
           and   | D_0 (y) | ≲exp (- γ | y
         |^θ),

  (see, i.e., <cit.> Section 1.2.2 Proposition 1 or
  <cit.>).



  In this appendix we treat only the case of polynomial
  weight. More generally all the statements here presented can be extended to
  the case of regular enough weights  ω satisfying the following
  conditions
  
    0 < ω (y)^± 1 < c ω (w - y)^± 1exp (d | w |)

  
    | D^αω (y) | ⩽ e ω (y)

  where c, d, e > 0 and y, w ∈ℝ^n. Here we have decided to
  present the case with polynomial weights because for finite distributions
  with respect to weights satisfying only the conditions (<ref>)
  and (<ref>) the Fourier transform is not defined. This means
  that we cannot use directly Definition <ref> and a more
  technical discussion is needed. We refer to <cit.> for the
  study of weighted Besov spaces with a weight satisfying the conditions
  (<ref>) and (<ref>).


We denote by 𝔉_r, with r ∈ℕ, the space of
functionsf ∈ C^r (ℝ^n) with support in B_1 (0) and norm f
_ C^r (ℝ^n)≤ 1. If f ∈𝔉_r we
write f_y, λ (·) = λ^- n f ( · -
y/λ), y ∈ℝ^n and λ > 0. 

  For any s < 0, p, q ∈ [1, + ∞]
  and ℓ∈ℝ an equivalent norm in the space B^s_p, q, ℓ
  is given by the following expression
  
    f _B^s_p, q, ℓ∼( ∫_0^1 sup_g ∈𝔉_r | ⟨ f, g_·, λ⟩ | _p,
        ℓ^q/λ^s qλ/λ)^1 / q .

  where r ∈ℕ is the first integer bigger than - s when s <
  0.



  Theorem 6.15, in <cit.>, proves the equivalence between norm
  (<ref>) and the norm of B^s_p, q, ℓ built using wavelets,
  while Proposition 2.4 in <cit.> proves the equivalence between
  the norm of B^s_p, q, ℓ built using wavelets and the norm
  (<ref>). Combining the two results we obtain the thesis.


If f is a measurable function and N ∈ℕ we define

    Δ^N_h (f) (y) = ∑_i = 0^N (- 1)^N - i( [ N; i ]) f (y + h i)

for x, h ∈ℝ^n.


  For any s > 0, p, q ⩾ 1, ℓ∈ℝ, and N, M ∈ℕ_0 such that N < s < M an equivalent
  norm in the space B^s_p, q, ℓ is given by the following expression
  
    f _B^s_p, q, ℓ∼ f _L^p_ℓ + ∑_α∈ℕ^n, | α | = N( ∫_| h | < 1 | h |^- (s - N) qΔ_h^M D^α f _L^p_ℓ^q  h/| h |^n)^1/q .




  The proof is given in <cit.> Theorem 6.9 for the case N = 0.
  The generic case N > 0 can be proved applying the techniques of Corollary
  2 Section 2.6.1 of <cit.> (where formula
  (<ref>) is proven for Besov spaces without weight).



  An important consequence of Proposition
  <ref> is that for any N ∈ℕ such that N
  < s we have that an equivalent norm of B^s_p, q, ℓ is given by
  
    f _B^s_p, q, ℓ∼ f _L^p_ℓ + ∑_α∈ℕ^n, | α | = N D^α f _B^s - N_p, q, ℓ
        .




  Consider p_1, p_2, q_1, q_2 ∈ [1, ∞],
  ℓ_1,ℓ_2 ∈ℝ and s_1, s_2 ∈ℝ such that
  s_1 - n/p_1 > s_2 - n/p_2 and ℓ_1 > ℓ_2 then
  B^s_2_p_2, q_2, ℓ_2⊂ B^s_1_p_1, q_1, ℓ_1 and the
  immersion is compact.
  
  Furthermore for 1 ≤ p ≤ 2
  
    B^s_p, p, ℓ⊂ W^s, p_ℓ⊂ B^s_p, 2, ℓ ;

  for 2 ≤ p < ∞
  
    B^s_p, 2, ℓ⊂ W^s, p_ℓ⊂ B^s_p, p, ℓ

  and for p = ∞ W^s, ∞_ℓ⊂ B^s_∞, ∞.
  Each of above immersions is continuous. 



  The proof of the first part of the proposition can be found in Theorem 6.7
  in <cit.>.
  
  The second part of the proposition is proved in Theorem 6.4.4 and Theorem
  6.2.4 of <cit.> for unweighted spaces. The result for weighted
  spaces follows from the fact that f ∈ B^s_p, q, ℓ and g ∈ W^s,
  p_ℓ if and only if f · r_ℓ∈ B^s_p, q and g ·
  r_ℓ∈ W^s, p (see <cit.> Theorem 6.5 point ii) ). 



  Consider p_1, p_2, p_3, q_1, q_2, q_3 ∈ [1,
  ∞] such that 1/p_1 + 1/p_2 = 1/p_3 and q_1
  = q_3 and q_2 = ∞. Moreover consider ℓ_1,ℓ_2, ℓ_3 ∈ℝ with ℓ_1 + ℓ_2 = ℓ_3 and consider s_1 < 0 s_2 ≥
  0 and s_3 = s_1 + s_2 > 0. Finally, consider the bilinear functional   Π (f, g) = f · g defined on 𝒮 (ℝ^n) taking
  values in 𝒮 (ℝ^n). Then there exists a unique
  continuous extension of Π as the map
  
    Π : B^s_1_p_1, q_1, ℓ_1× B^s_2_p_2, q_2, ℓ_2→ B^s_3_p_3, q_3, ℓ_3

  and we have, for any f, g for which the norms are defined:
  
    Π (f, g) _B^s_3_p_3, q_3, ℓ_3≲ f
         _B^s_1_p_1, q_1, ℓ_1 g _B^s_2_p_2, q_2, ℓ_2 .




  The proof can be found in <cit.> in Section 3.3 for Besov spaces with
  exponential weights. The proof for polynomial weights is similar.



  Consider p_1, p_2, q_1, q_2 ∈ [1,
  ∞], ℓ_1, ℓ_2 ∈ℝ and s_1, s_2 ∈ℝ. For
  any θ∈ [0, 1] we write θ/p_1 + 1 -
  θ/p_2 = 1/p_θ, θ/q_1 + 1 -
  θ/q_2 = 1/q_θ, θℓ_1 + (1 - θ) ℓ_2 =
  ℓ_θ and θ s_1 + (1 - θ) s_2 = s_θ. If f ∈
  B^s_1_p_1, q_1, ℓ_1∩ B^s_2_p_2, q_2, ℓ_2 then f ∈
  B^s_θ_p_θ, q_θ, ℓ_θ, and furthermore
  
    f _B^s_θ_p_θ, q_θ, ℓ_θ≤
         f _B^s_1_p_1, q_1, ℓ_1^θ f _B^s_2_p_2, q_2,
         ℓ_2^1 - θ .

  Furthermore if f ∈ W^s_1, p_1_ℓ_1∩ W^s_2, p_2_ℓ_2 then
   f ∈ W^s_θ, p_θ_ℓ_θ, and furthermore
  
    f _W^s_θ, p_θ_ℓ_θ≤ f
         _W^s_1, p_1_ℓ_1^θ f _W^s_2, p_2_ℓ_2^1 -
         θ .




  The proof is based on the fact that the complex interpolation
  (B^s_1_p_1, q_1, ℓ_1, B^s_2_p_2, q_2, ℓ_2)_θ of the
  two spaces B^s_1_p_1, q_1, ℓ_1, B^s_2_p_2, q_2, ℓ_2 is
  given by B^s_θ_p_θ, q_θ, ℓ_θ. This
  interpolation is proved in <cit.> Theorem 6.4.5 point (6) for
  unweighted space. For weighted space it follows from the fact that f ∈
  B^s_i_p_i, q_i, ℓ_i if and only if f · r_ℓ_i∈
  B^s_i_p_i, q_i.
  
  A similar reasoning, using the interpolation proved in <cit.>
  Theorem 6.4.5 point (7), holds also for the second inequality.


plain





































































































































































































































































































































































































