Another probabilistic construction of $Φ^{2n}$ in dimension 2
Yichao Huang
math.PR math-ph math.MP


A super scalable algorithm for short segment detection
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§ INTRODUCTION


The main input of this note is to provide an alternative probabilistic approach to the ϕ^2n theory in dimension 2, based on concentration phenomenon of martingales associated to polynomials of Gaussian variables. This is based on an adaptation of the work <cit.> of Lacoin-Rhodes-Vargas, in which exponential potentials associated to quantum Mabuchi K-energy are studied.

We give an alternative proof of the following classical result.

[Negative exponential moments]
Let n≥ 2 be an integer and let R be a real, unitary polynomial of even degree 2n. Let X be the (Dirichlet) Gaussian Free Field on a bounded simply connected domain Λ⊂ℝ^2.

Consider the (non-necessary positive) Wick-ordered random measure

    V_R(Λ)=∫_Λ:R(X)(x):d^2x

with integer n≥ 2. Then we have the following estimate

    𝔼[e^-α V_R(Λ)]<∞

for some α>0.


This key estimate for the construction of the ϕ^2n theory (where R(X)=X^2n) in dimension 2 follows originally from a hypercontractivity argument due to Nelson <cit.>. Given this estimate the rest of the argument is standard: the book <cit.> is a good reference for details and developments of the hypercontractivity argument. 

The idea of the martingale method is originally used to study more involved models such as the quantum Mabuchi K-energy <cit.> or the Sine-Gordon model <cit.>. This note shows in particular that this idea can also be sucessfully implemented to the Euclidean quantum ϕ^2n theory in dimension 2.

We stress that the purpose of this note is to introduce a new and arguably convenient construction of a classical theory in an elementary fashion. Readers unfamiliar with the classical model can consult <cit.> for an overview on this subject.



 §.§ Acknowledgement

The author is indebted to Christophe Garban, Rémi Rhodes and Vincent Vargas for communicating the idea. We also acknowledge support from the ERC grant QFPROBA.



§ PRELIMINARIES



  §.§.§ Notations
 In the following we denote by Λ⊂ℝ^2 some bounded simply connected open subset of the Euclidean plane ℝ^2. We consider n≥ 2 be an integer and let R be a real, unitary polynomial of even degree 2n. We use X to denote the (Dirichlet) Gaussian Free Field (GFF in short) supported on Λ. The object of interest would be the Wick-ordered polynomial :R(X): for the GFF. More precisely, we are interested in integrals of type ∫_Λ :R(X)(x): d^2x.



 §.§ Gaussian Free Field

We review some of the aspects of the probabilistic construction of the Gaussian Free Field (or GFF after) that will be useful later. We refer to <cit.> for more information.

Recall that the Green function K(x,y) on the domain Λ is defined as K=(-Δ_Λ)^-1, where -Δ_Λ is the differential operator with Dirichlet boundary condition g=0 on ∂Λ. In the following we will stick to the Dirichlet boundary condition although the argument works for general boundary conditions.

A (Dirichlet) GFF X on Λ is a random distribution taking value in the negative Sobolev space H^-s(Λ) with s>0. It is characterized by its mean and covariance kernel K on Λ: for test functions f,g∈ H^s(Λ),

    𝔼[⟨ X,f⟩]=0,  𝔼[⟨ X,f⟩⟨ X,g⟩]=∫_Λ^2f(x)g(y)K(x,y)d^2xd^2y

where ⟨,⟩ denotes the dual bracket between H^-s(Λ) and H^s(Λ). Recall that the Green function K displays logarithmic divergence on the diagonal, that is

    K(x,y)=-ln|x-y|+F(x,y)

with F(x,y) smooth.



 §.§ Wick ordering and Hermite polynomials

Let (B_t)_t∈ℝ_+ be the standard 1d Brownian motion. We consider the Wick ordering of (B_t)^2n, defined by

    :(B_t)^2n: = t^n P^H_2n(B_t/√(t))

where P^H_2n denotes the Hermite polynomial (normalized to have unitary leading coefficient) of degree 2n. The Wick ordering procedure requires that the expectation vanishes, i.e.

    𝔼[P^H_2n(B_t/√(t))]=0,  ∀ t≥ 0.

It follows that the Itô derivative of P^H_2n(B_t) with respect to the Brownian filtration has no drift term. The Wick ordering procedure provides a natural martingale parametrized by the time t.

Notation. In the following we absorb the renormalization in t for P^H_2n and write

    P_2n(B_t) t^n P^H_2n(B_t/√(t)).


Example. For n=2, the Wick ordering yields

    P_4(B_t)=B_t^4-6t B_t^2+3t^2

which can be equally written as

    P_4(B_t)=(B_t^2-3t)^2-6t^2

and P_4 is bounded from below by -6t^2. We also deduce that the envelope of the zero-graph

    {(t,B_t)∈ℝ_+×ℝ, P_2n(B_t)=0}

is given by two symmetric branches

    ⋃_t∈ℝ_+{(t,√((3+√(6))t))}∪{(t,-√((3+√(6))t))}⊂ℝ_+×ℝ.


General case.  In general, by linear combination, we define the Wick ordered polynomial of B_t for any real, unitary polynomial R of even degree 2n≥ 4:

    :R(B_t):=P_R(B_t).

More precisely, if

    R(X)=∑_i=0^2na_iX^i

with a_2n=1, then we define the associated Wick ordered polynomial P_R(X) by

    P_R(X)=∑_i=0^2na_it^i/2P^H_i(X/√(t)).

The martingale property of P_R(B_t) with respect to the Brownian filtration is preserved by linear combination. The envelope of the graph of the zeros of P_R is given explicitly by

    ⋃_t∈ℝ_+{(t,f_R(t))}∪{(t,-f_R(t))}⊂ℝ_+×ℝ

where the positive branch f_R≥ 0 can be explicitly calculated. The example above shows that when R(X)=X^4,

    f_X^4(t)=√((3+√(6))t).


The following facts are elementary.

Let R be a real, unitary polynomial of even degree 2n≥ 4. The function f_R satisfies the following:

	
  * There exists some constant A>0 only depending on n such that f_R(t)≤ t+A for all t∈ℝ_+;
	
  * For every ϵ>0, there exists some constant A'=A'(n,ϵ) such that f_R(t)≤ϵ t+ A' for all t∈ℝ_+.



We will also consider the value of P_R on the line {t+A}_t≥ 0 for some constant A.

For large enough A, the function

    t↦ P_R(t+A)

satisfies the following properties:

	
  * It is positive for t∈ℝ_+;
	
  * It is strictly increasing in t for t∈ℝ_+.





 §.§ Cut-off regularization

Since the Gaussian Free Field X only makes sense as a distribution, it is suitable to define the measure

    V_R(Λ)=∫_Λ:R(X)(x):d^2x

using a cut-off procedure. We need the following assumption:

We choose a cut-off regularization (X_u)_u∈ℝ_+ satisfying the following properties:

	
  * The covariance kernel K can be written in the form
	
    K(x,y)=∫_0^∞Q_u(x,y)du

	where for all x≠ y, the above integral is convergent; Q_u is a bounded symmetric positive definite kernel for any u.
	
  * Setting K_t=∫_0^tQ_u du, there exists a positive constant C such that
	
    |K_t(x,y)-(t∧ln_+1/|x-y|)|≤ C.

	
  * We have lim_x→∞Q_u(x,x)=1 with uniform convergence in x∈Λ.
	
  * For all 0<β<2,
	
    ∫_Λ^2∫_0^∞e^β u|Q_u(x,y)|d^2x d^2y du<∞.



It is proven in <cit.> that the GFF X on Λ can be fitted into this assumption. We will thus work under this assumption in the following.

We define (X_t(x))_x∈Λ,t≥ 0 to be the jointly continuous process in x and t with covariance kernel

    𝔼[X_s(x)X_t(y)]=∫_0^t∧ sQ_u(x,y)du.

According to the above assumption, given x∈Λ, the process (X_t(x))_t≥ 0 is very similar to a standard Brownian motion. We assume for readability in the following that

    K_t(x,x)=t

so that (X_t(x))_t≥ 0 is a standard Brownian motion.



 §.§ Quadratic variation of martingales

We have the following lemma in probability concerning the exponential martingale:
[Exponential martingale]
For any continuous local martingale M and any λ∈ℂ, the process

    exp(λ M_t-λ^2/2<M,M>_t)

is a local martingale. We also write <M>_t for the quadratic variation <M,M>_t.


In particular, if M_0=1 and <M>_∞<∞, then M is a L^2-bounded continuous martingale and we have for α>0 the following inequality

    lim sup_t→∞𝔼[exp(-α M_t)]<∞

in such a way that the limit of M_t displays Gaussian concentration.

We refer to the classical text book <cit.> for these results.














§ PROOF OF THE MAIN THEOREM

We now prove Theorem <ref> using martingale methods.



 §.§ Preliminary notations

Fix a real, unitary polynomial R of even degree 2n≥ 4. Hereafter we sometimes drop the dependence on R where there is no ambiguity.

Borrowing notations from Proposition <ref>, we consider the two-branched envelope

    E{(t,u)∈ℝ_+×ℝ;|u|=f_R(t)}.

The envelop E depends on R: we write simply E for readability.

We introduce a cut-off at level ± g(t) where

    g(t)=t+A

and A≥ 0 is a large constant chosen later: we require that f_R(t)≤ g(t) for all t∈ℝ_+ and that Proposition <ref> holds.

We consider also the cone C with two symmetric branches:

    C{(t,u)∈ℝ_+×ℝ;|u|=g(t)}.

Geometrically, the envelope E is in between the two branches of C.

Let us rewrite the main theorem with the notations in the preliminary. We define a regularization of the measure V_R(Λ) using the smooth white noise decomposition Proposition <ref>:

    D_t=∫_ΛP_R(X_t(x))d^2x.

We prove in the following that uniformly in t, there exists some C(A) such that for all α∈ℝ,

    𝔼[e^α D_t]≤ e^C(A)α^2.




 §.§ Strategy of the proof

One first calculates the quadratic variation of the martingale D_t in view of Lemma <ref>. We have

    <D>_t≤∫_Λ^2×[0,t]|P'_R(X_u(x))P'_R(X_u(y))|Q_u(x,y)d^2x d^2y du.

If the graph (t,X_t(x)) stays (uniformaly in t and in x) inside the cone C, then |P'_R(X_u(x))| cannot take exceptionally high values and the quadratic variation <D>_t is uniformaly bounded in t (see Lemma <ref> below) and the L^2-theory of martingales applies. By Lemma <ref>, the limiting measure would display Gaussian concentration bound.

Almost surely this is not the case: the process X_t(x) goes out of the cone C and takes high values. We consider for every x∈Λ the stopping time

    H^xinf{s≥ 0;(s,X_s(x))∈ C}.

As the zero-value envelope E is inside the cone C, after time H^x the process P_R(X_t(x)) at point x stays positive until the next time it returns to E.

Introduce a sequence of stopping times (always with respect to a fixed x∈Λ):

    H^x_k   inf{s≥ L^x_k-1;(x,X_s(x))∈ C},
    
    L^x_k   inf{s≥ H^x_k;(x,X_s(x))∈ E}.

By convention, L^x_0≡ 0. We can write as a decomposition of times depending on whether X_t(x) takes low or high values,

    [L^x,H^x]   ⋃_k∈ℕ[L^x_k,H^x_k+1],
    
    [H^x,L^x]   ⋃_k∈ℕ^*[H^x_k,L^x_k].

It follows that for all x∈Λ, [L^x,H^x]∪[H^x,L^x]=ℝ_+ almost surely.

We will take advantage of the positivity between the stopping times [H^x,L^x]. More precisely, on one hand the total contribution of P_R(X_t(x)) from intervals of the form [L^x,H^x] is bounded in L^2 (since it takes values inside the cone C), on the other hand the contribution of P_R(X_t(x)) from intervals [H^x,L^x] has constant positive sign.

We quantify this observation in the following way:

We consider the following decomposition. Let

    D_L(t)=∫_Λ(∫_0^tP'_R(X_s(x))1_{s∈[L^x,H^x]}ds)d^2x

and

    D_H(t)=∫_Λ(∫_0^tP'_R(X_s(x))1_{s∈[H^x,L^x]}ds)d^2x

in such a way that

    D_t=D_L(t)+D_H(t).

Then we have the following inequality

    D_t≥ D_L(t)-Q

where Q denotes the positive quantity

    Q∫_Λ(∑_i=1^∞1_{H^x_i<∞}P_R(g(H^x_i)))d^2x.

Note that D_L, D_H, Q depend on R but we drop this dependence in the notation.


Fix x∈Λ and one can check the following claims:

	
  * If k∈ℕ is such that t∈[L^x_k,H^x_k+1], then

    ∫_0^tP'_R(X_s(x))1_{s∈[L^x,H^x]}ds
       =(P_R(X_t(x))-P_R(X_L^x_k(x)))+∑_i=0^k-1(P_R(X_H^x_i+1(x))-P_R(X_L^x_i(x))).

This is because for every l<k, the increment of the process X_t(x) on the interval [L^x_l,H^x_l+1] contributes exactly to one term in the above summation.
	
  * If now k∈ℕ is such that t∈[H^x_k,L^x_k], then we have P_R(X_t(x))≥ 0 and

    ∫_0^tP'_R(X_s(x))1_{s∈[L^x,H^x]}ds=∑_i=0^k-1(P_R(X_H^x_i+1(x))-P_R(X_L^x_i(x))).



Notice now that for all i∈ℕ, P_R(X_L^x_i(x))=0 by definition of the zero envelope E and hitting times L^x_i. A similar argument as above shows that D_H(t)≤ 0 for all t∈ℝ_+, so that

    D_t≥ D_L(t).


To prove Equation (<ref>), write the above in the following form:

    P_R(X_t(x))≥∫_0^tP'_R(X_s(x))1_{s∈[L^x,H^x]}ds-∑_i=0^∞1_{H^x_i<∞}P_R(X_H^x_i+1(x)).

Equation (<ref>) follows by integrating over x∈Λ.


Now the proof of the main theorem boils down to two estimates, of which the first one corresponds to the L^2 part, and the second one corresponds to the high-value part.
[Low value contribution]
D_L(t) is an honest martingale that has bounded quadratic variation: it converges in L^2 and satisfies the Gaussian concentration bound

    ∃ C(A), ∀α∈ℝ, 𝔼[e^α D_L(∞)]≤ e^C(A)α^2.


[High value contribution]
The other quantity Q in the decomposition also satisfies a Gaussian concentration bound:

    ∃ C(A), ∀α∈ℝ, 𝔼[e^α Q]≤ e^C(A)α^2.


Combining these two lemmas, Theorem <ref> follows.



 §.§ Proofs of technical estimates

We start by proving Lemma <ref>.

The fact that D_L(t) is a martingale follows from construction. It suffices to show that <D_L>_∞ is bounded from above by a constant: Gaussian concentration then follows by Lemma <ref>. The calculation goes as follows:

    <D_L>_t≤∫_Λ^2×[0,t]|P'_R(X_s(x))P'_R(X_s(y))|1_{s∈[L^x,H^x]∩[L^y,H^y]}Q_u(x,y)d^2x d^2y du.


Since P'_R(X_s(x)) is polynomial of degree 2n-1, it has subexponential growth at infinity and the conditioning on s implies that |X_s(x)|≤ g(s). We bound the above by

    <D_L>_t   ≤ C∫_Λ^2×[0,t]e^1/2g(s)e^1/2g(s)1_{s∈[L^x,H^x]∩[L^y,H^y]}Q_s(x,y)d^2x d^2y ds
       ≤ C∫_Λ^2×[0,t]e^sQ_s(x,y)d^2x d^2y ds

for some constant C=C(R). The last integral is finite by the last item of Proposition <ref>.



Recall some preliminaries on Doob martingales. Define the positive quantity

    Q^x=∑_i=1^∞1_{H^x_i<∞}P_R(g(H^x_i))

(Q^x depends on R but we alleviate the notation) so that

    Q=∫_ΛQ^x d^2x.

[ℒ^1-boundedness]
We have 𝔼[Q]<∞.


We bound 𝔼[Q^x] uniformly in x∈Λ: the claim follows from integrating over Λ.

Consider the following quantity:

    Q^x,m=∑_i=1^∞1_{H^x_i∈(m-1,m]}P_R(g(m)).

By Proposition <ref>, choose A large enough such that P_R is strictly increasing on ℝ_+ and

    P_R(g(m))=sup_v∈(m-1,m]P_R(g(v))

such that

    Q^x≤∑_m=1^∞Q^x,m.


We now prove a standard estimate

    𝔼[#{i:H^x_i∈(m-1,m]}]≤8/√(2π m)e^-m/2.

Given this and that the polynomial P_R has sub-exponential growth at infinity, i.e.

    P_R(g(m))≤ C(A)e^m/4,

the result follows by summing over m then integrating over x. 

Notice that, with A≥ 1, 

    ℙ[∃ i, H^x_i∈(m-1,m]]≤ℙ[sup_s≤ m|B_s|≥ m]≤4/√(2π m)e^-m/2

by a standard Gaussian tail estimate. Using the Markov property for the Brownian motion,

    ℙ[#{i:H_i∈(m-1,m]≥ k+1}|#{i:H_i∈(m-1,m]≥ k}]≤1/2

and Equation (<ref>) follows from summing over k.


The Doob martingale Q^x_t is defined as

    Q^x_t=𝔼[Q^x|ℱ_t]

(recall that ℱ_t=σ{X_s,s∈[0,t]}) and since it is a martingale associated to the Brownian filtration {X_t(x)}, we can write

    dQ^x_t=A^x_t dX_t(x).

Then the bracket <Q>_∞ can be written as

    <Q>_∞=∫_Λ^2×ℝ_+A^x_u A^y_u Q_u(x,y)d^2x d^2y du.


We now control A^x_t uniformly in x, according to whether t∈[H^x,L^x] or t∈[L^x,H^x]. In the following we drop the dependency on x to alleviate the notations. More precisely, we prove that uniformly over all t, with some constant C(A) independent of x,

    |A_t|≤ C(A)e^t/2.

Lemma <ref> then follows from Equation (<ref>) and Lemma <ref>, together with Lemma <ref>.

To prove Equation (<ref>), we apply coupling techniques to the Brownian motion X_t(x).



  §.§.§ First case: t∈[H^x,L^x]

Suppose t∈[H^x_k,L^x_k] for some k∈ℕ.

Let ℙ_z be the law of a standard Brownian motion (B_t)_t≥ 0 starting at point z. By the strong Markov property of X_t(x) as a Brownian motion B_t (we drop the index x afterwards), write Q_t as

    Q_t=∑_i=1^k-1P_R(g(H_i))+𝔼_X_t(x)[∑_i=1^∞1_{H^t_i<∞}P_R(g(t+H^t_i))]

where the stopping time sequence H^t_k=H^t_k(B) is defined recursively by H^t_0=0 and

    L^t_k   =inf{s≥H^t_k-1; B_s∈ E};
    H^t_k   =inf{s≥L^t_k; B_s∈ C}.

We deduce the expression for A_t in this case:

    A_t=∂_z(𝔼_z[∑_i=1^∞1_{H^t_i<∞}P_R(g(t+H^t_i))])|_z=X_t(x).

We show that the expression in the definition of A_t that we derive is Lipschitz in z with the adequate Lipschitz constant: this will imply Equation (<ref>).

Let t∈[H,L] and consider a coupling between two independent Brownian motions, starting from points z_1<z_2 with |z_1-z_2| small, denoted respectively by B^1 and B^2. Suppose that the two Brownian motions evolve independently until the first time they meet

    τ=inf{s>0; B^1_s=B^2_s}

and jointly afterwards. Each Brownian motion in this coupling defines its own hitting time H^(t,j)_i,L^(t,j)_i for j∈{1,2} similarly as in Equations (<ref>). The hitting times are identical up to a shift in the indices after merging at time τ.

If τ<min{H^(t,1)_1,H^(t,2)_1}, then each Brownian motion gives rise to the same contribution in the expression of A_t. In particular, this also holds for τ<min{H^(t,1)_1,H^(t,2)_1,1}. Hereafter let

    𝒯=min{H^(t,1)_1,H^(t,2)_1,1}.


It suffices to show that the following bound:

    |𝔼_z_1[∑_i=1^∞1_{H^t_i<∞}P_R(g(t+H^t_i))]-𝔼_z_2[∑_i=1^∞1_{H^t_i<∞}P_R(g(t+H^t_i))]|
       ≤ℙ[τ>𝒯]×𝔼[∑_j=1,2∑_i=1^∞1_{H^(t,j)_i<∞}P_R(g(t+H^(t,j)_i))|τ>𝒯].


∙ One first shows by standard coupling estimate on Brownian motions that

    ℙ[τ>min{H^(t,1)_1,H^(t,2)_1,1}]≤ C|z_1-z_2|.

It is a standard Brownian coupling result that ℙ[τ>1]≤ C|z_1-z_2|. It remains to show

    ℙ[τ>H^(t,1)_1]≤ C|z_1-z_2|.

Provided that we choose a large enough A in the definition of g_t, we have

    H^(t,1)_1≥min{s;|B^1_s-z_1|≥ 1}

and

    ℙ[τ>H^(t,1)_1]≤ℙ_(0,z_2-z_1)[𝒯_Δ>𝒯_{-1,1}×ℝ]≤ C|z_1-z_2|

where Δ={(x,x);x∈ℝ} and 𝒯_A denotes the hitting time of a set A by a two-dimensional Brownian motion. This is a standard estimate (for a detailed proof, <cit.>).

∙ We now show that

    𝔼[∑_j=1,2∑_i=1^∞1_{H^(t,j)_i<∞}P_R(g(t+H^(t,j)_i))|τ>min{H^(t,1)_1,H^(t,2)_1,1}]≤ Ce^t/2.


By linearity it suffices to show it for j=1, the calculation for j=2 is similar. We apply Markov property at 𝒯=min{H^(t,1)_1,H^(t,2)_1,1} and distinguish two subcases:

– If 𝒯<L^(t,1)_1, we apply Markov property for the Brownian motion B^1_s at L^(t,1)_1. Since at time L^(t,1)_1 the Brownian motion B^1 takes value f_R(t+L^(t,1)_1), the above quantity is dominated by

    sup_r∈[t,t+1]sup_s≥ r𝔼_f_R(s)[∑_i=1^∞1_{H^(s,1)_i<∞}P_R(g(s+H^(s,1)_i))].

As in Lemma <ref>, together with Proposition <ref> for any r∈[t,t+1] and s≥ r,

    𝔼_f_R(s)[∑_i=1^∞1_{H^(s,1)_i<∞}P_R(g(s+H^(s,1)_i))]
       ≤ C∑_n≥ 1𝔼_f_R(s)[#{H^(s,1)_i∈(n-1,n]}]P_R(g(s+n))
    
       ≤ C∑_n≥ 11/√(n)e^-((1-ϵ)s+n)^2/2ne^(s+n)/3
       ≤ Ce^-s/3.

We used the Proposition <ref> that one can assume f_R(s)<ϵ s+A'(ϵ) uniformly for any ϵ>0.

– If 𝒯>L^(t,1)_1, then since 𝒯≤H^(t,1)_1 by definition, we know that

    |B^1_𝒯|≤ g(t+𝒯).

By assumption, 𝒯≤ 1 and we can control the contribution by

    sup_r∈[t,t+1]sup_|z|≤ g(r)𝔼_z[∑_i=1^∞1_{H^(r,1)_i<∞}P_R(g(r+H^(r,1)_i))].

Again, by the same argument as in Lemma <ref>, with |z|≤ g(r) and Proposition <ref>,

    𝔼_z[∑_i=1^∞1_{H^(r,1)_i<∞}P_R(g(r+H^(r,1)_i))]
       ≤ C∑_n≥ 1ℙ_z[∃ i; H^(r,1)_i∈(n-1,n]]P_R(g(r+n))
       ≤ C∑_n≥ 11/√(n)e^-n/2e^(r+n)/3
       ≤ Ce^r/3

so that the contribution above is control by Ce^t/3. This is an appropriate Lipschitz constant for Equation (<ref>).





  §.§.§ Second case: t∈[L^x,H^x]

In this case, the above strategy fails for the first term i=1. Indeed, if both Brownian motions start near the cone C, the probability that they merge before either of them hitting C is arbitrarily small and Equation (<ref>) cannot be reproduced. However the same argument works for i≥ 2 (since in this case they both have to travel from the inner envelope E to the outer cone C). We thus have to look more carefully into the term i=1.

For the i=1 case, we use a different “parallel” coupling. Consider two Brownian motions, B^1_s starting at z_1 and B^2_s starting at z_2 (by symmetry, suppose that z_2>z_1) coupled as

    B^2_s=B^1_s+(z_2-z_1).


Denote by S_1 (resp. S_2) the hitting time of B^1 (resp. B^2) at the outer cone C. We show that

    |𝔼[1_{S_1<∞}P_R(g(t+S_1))]-𝔼[1_{S_2<∞}P_R(g(t+S_2))]|≤ C(A)|z_1-z_2|e^t/2.


By symmetry we can add the indicator of the event that S_1<S_2 (otherwise change (z_1,z_2) into (-z_1,-z_2)). This is only a geometric data: with the assumption that z_1<z_2, the event S_1<S_2 is equivalent to the event that the first time any of the Brownian motions B^1_s and B^2_s hits the the outer cone C, the location is at the lower branch of C.

Since the above inequality is an absolute value, we should seperate into two subcases:

∙ We can choose A large enough so that P_R(g(·)) is strictly increasing on ℝ_+ by Proposition <ref>. Notice that with the conditioning S_1<S_2 and Markov property at S_1,

    𝔼[1_{S_2<∞}P_R(g(t+S_2))1_{S_1<S_2}]
       ≥𝔼[1_{S_2<∞}P_R(g(t+S_1))1_{S_1<S_2}]
       ≥𝔼[1_{S_1<∞}P_R(g(t+S_1))1_{S_1<S_2}]inf_S_1>0ℙ[S_2<∞|S_1<∞]

and thus we have

    𝔼[1_{S_1<∞}P_R(g(t+S_1))1_{S_1<S_2}]-𝔼[1_{S_2<∞}P_R(g(t+S_2))1_{S_1<S_2}]
       ≤(sup_S_1>0ℙ[S_2=∞|S_1<∞])𝔼[1_{S_1<∞}P_R(g(t+S_1))1_{S_1<S_2}].


Repeating arguments as before, we have

    𝔼[1_{S_1<∞}P_R(g(t+S_1))1_{S_1<S_2}]≤𝔼[1_{S_1<∞}P_R(g(t+S_1))]≤ C(A)e^t/2.


It remains to control the other conditional probability. Using the strong Markov property for B^2 at S_1, for all S_1>0,

    ℙ[S_2<∞|S_1<∞]≥ℙ_z_1-z_2[∃ s; B_s=s]=e^-2(z_2-z_1)

by standard diffusion process identity. Indeed, for z≤ 0,

    u(z)=ℙ_z[∃ s; B_s=s]

solves the differential equation

    u”(z)-2u'(z)=0

with initial condition u(0)=1 and u(-∞)=0. Together this proves one direction in Equation (<ref>)

∙ Now we prove the other direction. Let 𝒯=min{S_2,S'_2} with

    S'_2=inf{s>S_1;B^2_s=-f_R(t+s)}.

Only the lower branch of the envelope E is concerned because the assumptions z_1<z_2 and S_1<S_2 imply geometrically that at time S_1, the Brownian motion B^1_S_1 is located at the lower part of the cone C and the other coupled Brownian motion B^2_S_1, at time S_1, is between the lower part of E and the lower part of C. It follows that after time S_1, the Brownian motion B^2 first hits either the lower part of E (corresponding to S'_2) or the lower part of C (corresponding to S_2).







We give bounds on 𝔼[1_{S_2<∞}P_R(g(t+S_2))1_{S_1<S_2}] depending on how S_2 compares to S'_2. We are going to show that in one case

    𝔼[1_{S_2<∞}P_R(g(t+S_2))1_{S_1<S_2}1_{S_2<S'_2<∞}]≤𝔼[1_{S_1<∞}P_R(g(t+S_1))1_{S_1<S_2}]

and in the other case

    𝔼[1_{S_2<∞}P_R(g(t+S_2))1_{S_1<S_2}1_{S'_2<S_2<∞}]≤ C|z_1-z_2|.

The sum of these two equalities yields a constant order Lipschitz coefficient for Equation (<ref>).

– For the first case, consider the Brownian motion

    B_s=B^2_S_1+s-B^2_S_1.

Now in this case, 𝒯-S_1 is a stopping time for B. Since conditioned on the event that S_1<∞ and S_1<S_2, {P_R(B^2_S_1+B_u)}_u≥ 0 is a (positive) martinagle for the filtration

    ℱ_u=ℱ_S_1∪σ(B_s;s≤ u),

up until time (𝒯-S_1). Fatou's lemma for the conditional expectation yields (since P_R(f_R(·))=0)

    𝔼[1_{S_2<∞}P_R(g(t+S_2))1_{S_1<S_2}1_{S_2<S'_2<∞}]
       ≤𝔼[1_{S_1<∞}P_R(B^2_S_1)1_{S_1<S_2}]
       ≤𝔼[1_{S_1<∞}P_R(g(t+S_1))1_{S_1<S_2}]

provided that P_R(B^2_S_1)≤ P_R(g(t+S_1)) by Proposition <ref>.

– For the second case, consider

    𝔼[P_R(g(t+S_2))1_{S_1<S'_2<S_2<∞}].

By applying Markov property at time S'_2,

    𝔼[P_R(g(t+S_2))1_{S_1<S'_2<S_2<∞}]≤ℙ[S'_2<S_2<∞]max_s≥ t𝔼_f_R(s)[P_R(g(H^s_1))1_{H^s_1<∞}].


The second term on the right hand side is bounded by a constant, see Equation (<ref>). The rest reduces to the estimate (by applying Markov property at time S'_2)

    ℙ[S'_2<S_2|S_1<∞]
       =ℙ_z_1-z_2[min{s;B_s=f_R(s+t)-t-A}<min{s;B_s=s}]
       ≤ℙ_z_1-z_2[min{s;B_s=ϵ(s+t)+A'(ϵ)-t-A}<min{s;B_s=s}]

where we used the fact that f_R(s)≤ϵ s+A'(ϵ) for arbitrarily small ϵ>0. The last probability can be shown to be smaller than C|z_1-z_2| by diffusion process estimate. Indeed, it can be bounded by

    u(z_1-z_2)=ℙ_z_1-z_2[min{s;B_s=s+A'-A}<min{s;B_s=s}]

where the last term solves the differential equation

    u”(z)-2u'(z)=0

with initial conditions u(0)=0, u(A'-A)=1. Computation yields

    u(z)=1-e^-2z/1-e^-2(A'-A)≤ C|z|

which completes the proof.














































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