Wave equation with a coloured stable noise
Larysa Pryhara Georgiy Shevchenko
math.PR 60H15, 35L05, 35R60, 60G52





    M. I. Krivoruchenko^1,2,3
    December 30, 2023
=============================







§ INTRODUCTION


Being a common tools to model complex systems with randomness, stochastic partial differential equations have attracted a considerable attention. The number of articles in this area is constantly growing, and we mention only few of them which consider subjects closely related to our work; further references are available therein. The sample properties of wave equations with Gaussian random noise were studied in <cit.>. Several authors investigated equation with square integrable Lévy noise, where the same martingale methods are applicable, see <cit.> and references therein. 

However, in the heavy-tailed situation, there are only few results. Articles <cit.> are devoted to heat equations with stable noise. To the best of our knowledge, there are no articles studying the wave equation with a heavy-tailed noise, neither there are attempts to consider equations with coloured stable noise.  It is worth to mention that some authors investigated stochastic partial differential equations with a general stochastic measure, where, in particular, no assumptions on the integrability of measure are made. For example, the heat equation with a general stochastic measure was considered in <cit.>, the wave equation, in <cit.>. However, in these works only few results are proved and in rather restricted cases; it is probably the generality of exposition what leads to such restrictions. 

In this article we continue our research started in <cit.>, where a planar wave equation was studied, and <cit.>, which, as the present work, was devoted to wave equation in ^3. The main object of our study is the wave equation

    {(   ∂ ^2/∂ t^2- a^2 Δ)U(x,t)=Ż^H(x), x∈^3, t>0,
       U(x,0)=0, 
       ∂ U/∂ t(x,0)= 0,
    .

where the source is a spatial random noise with symmetric α-stable distribution: it is a derivative Ż^H(x) of a real anisotropic harmonizable fractional stable field Z^H. Thus, the random noise is “coloured” in the sense that its increments are not independent. To give a meaning to this equation requires defining a random measure generated by Z^H and an integral with respect to this measure. As far as we know, such questions did not appear in the previous literature. 

The rest of article is organized as follows. Section 2 contains preliminary information on stable random variables, measures and integrals. In Section 3, we recall the definition and properties of real anisotropic harmonizable fractional stable field, construct a random measure generated by this field, define integrals with respect to this measure and study their properties. In Section 4 we prove results on the sample path properties of the solution to (<ref>). 



§ PRELIMINARIES

Throughout the paper, C will denote a generic constant; its value may vary between lines. Random constants will be denoted by C(ω).

In this article we will consider symmetric α-stable (S α S) random variables. In this section we  will provide essential information about them; additional detail may be found in  <cit.>. 

For α∈(0,2), a random variable ξ is called Sα S with scale parameter ||ξ||_α if its characteristic function is 

    [e^iλξ]=e^-λ||ξ||_α^α, λ∈.


A crucial role in the construction of processes and fields with stable distribution is played by independently scattered Sα S random measure; in this article it is enough to consider a measure on ^3. This is a function M: ℬ_f(^3)×Ω→, where ℬ_f(^3) is a family of Borel sets of finite Lebesgue measure, having the following properties:

	
  * for any A ∈ℬ_f(^3), the random variable M (A) is  Sα S  with scale parameter equal to λ(A) the Lebesgue measure of A;
	
  * for any disjoint sets A_1, … ,A_n ∈ℬ_f(^3), the values M(A_1),… ,M(A_n)  are independent;
	
  * for any disjoint sets A_1, A_2,…∈ℬ_f(^3) such that ⋃_n=1^∞ A_n∈ℬ_f(^3), the series ∑_n=1^∞ M(A_n) converges almost surely and M(⋃_n=1^∞ A_n)=∑_n=1^∞ M(A_n) almost surely.

For a function f ∈ L^α(^3), the integral 

    I(f)=_^3f(x)M(dx)

is defined as a limit in probability of integrals of finitely supported simple functions, and there is an isometric property:

    I(f)^α_α=_^3|f(x,t)|^α dx.


A convenient tool to study stable random variables is the LePage series representation, defined for the measure M as follows. Let φ be any positive continuous probability density function on ^3, and {Γ_k,k≥1}, {ξ_k,k≥1}, {g_k,k≥1} be independent families of random variables satisfying

	
  * {Γ_k,k≥1} is a sequence of arrivals of a Poisson process with unit intensity;
	
  * {ξ_k,k≥1} are iid vectors in ^3 with density φ;
	
  * {g_k,k≥1} are iid centered Gaussian variables with  [|g_k|^α]=1.

Then {M(A),A∈ℬ_f(^3)} has the same distribution as 

    M'(A)=C_α∑_k≥1Γ_k^-1/αφ(ξ_k)^-1/α1_A(ξ_k)g_k, A∈ℬ_f(^3),

where C_α=(Γ(2-α)cosπα/2/1-α)^1/α; the series converges  almost surely for any A∈ℬ_f(^3). 
There is also the LePage series representation for integrals: for any f_1,f_2,…, f_n ∈ L^α(^3) the vector (I(f_1), I(f_2), … , I(f_n)) has the same distribution as (I'(f_1), I'(f_2), … , I'(f_n)), where

    I'(f)=C_α∑_k≥1Γ_k^-1/αφ(ξ_k)^-1/α f(ξ_k)g_k.

 
Further we will assume without loss of generality that M is given by  (<ref>) with 

    φ(x)=∏_l=1^3K/|x_l| (|log|x_l| |+1)^1+η,

where η>0 is a fixed number, and
K=(∫_-∞^+∞|x|^-1(|log|x| |+1)^-1-ηdx)^-1 is the normalizing constant.
Respectively, the integral I(f)=_^3f(x)M(dx) is given by (<ref>).

We will need several auxiliary results. For convenience, we reformulate them to suit our needs.
[<cit.>]
Let X_n = I(f_n) with some f_n∈ L^α(^3), n≥ 1. Then 

    I(f_n)𝖯⟶ I(f), n→∞,

if and only if _^3 |f_n(x) - f(x)|^α dx→ 0, n→∞. 




Assume that α∈ (1,2). For  𝐓∈ℬ (^m) and measurable  f_t(x): 𝐓×^3→ 
define

    X(t)=_^3f_t(x)M(dx),   t∈𝐓.

If

    ∫_𝐓|X(t)|dt<∞

almost surely, then

    ∫_𝐓X(t)dt=_^3∫_𝐓f_t(x)dt  M(dx)

almost surely.
A sufficient condition for (<ref>) to hold is

    ∫_𝐓(_^3|f_t(x)|^α dx)^1/αdt<∞.



This is a compilation of Theorem 3.3 and Theorem 4.1 from <cit.>. The measurability of X, assumed in these theorems, in our case is a consequence of the series representation <ref>.





§ REAL ANISOTROPIC HARMONIZABLE FRACTIONAL STABLE FIELD


The real anisotropic harmonizable fractional stable field with Hurst parameter H∈(0,1) is defined as

    Z^H(x)=_^3∏_l=1^3e^ix_ly_l-1/|y_l|^H+1/αM(dy)

for technical reasons we will restrict ourselves to the case α∈(1,2), H∈ (1/2,1).
The properties of Z^H were established in <cit.>, in <cit.> its multifractional was defined and investigated. In particular, it was proved in the cited articles that this field has a modification, which is locally Hölder continuous of any order β∈ (0,H).
Thanks to our assumptions about M, Z^H can be represented as its LePage series: 

    Z^H(x)=C_α∑_k=1^∞Γ_k^-1/α∏_l=1^3e^ix_lξ_k,l-1/|ξ_k,l|^H+1/αφ(ξ_k)^-1/αg_k,

 

    φ(x)=∏_l=1^3K/|x_l| (|log|x_l| |+1)^1+η.



Our aim is to define a random measure corresponding to Z^H(x). Towards this end, we will use the following heuristic reasoning. Differentiate  Z^H(x) informally:

    ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H(x)=(-i_^3e^i(x,y)∏_l=1^3 y_l/|y_l|^H+1/α-1M(dy))
    
    =_^3sin(x,y)∏_l=1^3 y_l/|y_l|^H+1/α-1M(dy).


Note that the integral is not well defined in general, but it allows to define the random measure corresponding to Z^H as a result of formal differentiation and change of order of integration:

    Z^H(A)=_A∂^3/∂ x_1∂ x_2 ∂ x_3Z^H(x) dx
    
    =_A_^3sin(x,y)∏_l=1^3 y_l /|y_l|^H+1/α-1M(dy)  dx
    
    =_^3∏_l=1^3 y_l /|y_l|^H+1/α-1_Asin(x,y) dy  M(dz).

In other words, we set

    Z^H(A)=_^3_Asin(x,y)dx∏_l=1^3 y_l /|y_l|^H+1/α-1M(dy)

by definition.

 
For any A∈ B_f(^3), the integral in (<ref>) is well defined.
 

Denote

    f_A(y)=_Asin(x,y)dx∏_l=1^3 y_l /|y_l|^H+1/α-1.

We need to show that 

    _^3|f_A(y)|^α dy <∞.

Note that f_A(y)=A^H1_A(y), where A^H is defined in (<ref>). Since 1_A(y) ∈ L^1(^3)∩ L^2(^3), the finiteness of integral follows from Proposition <ref>. 


In <cit.> we have proposed to define the integral with respect to Z^H as follows. For ε>0 define smooth approximations of Z^H by

    Z^H,ε(x)=_^3sin(x,y)e^-ε^2|y|^2/2∏_l=1^3 y_k,l/|y_k,l|^H+1/α-1M(dy).

It was proved in <cit.> that there exists a weak derivative ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H, ε(x) and, in view of this,  the following definition was proposed:

    _^3f(x)Z^H(dx)=lim_ε→ 0+_^3f(x)∂^3/∂ x_1∂ x_2 ∂ x_3Z^H,ε(x) dx,

provided that the limit in probability exists. Let us show that (<ref>) agrees with this definition.

For any A∈ℬ_f(^3), the following convergence in probability holds:

    Z^H(A)=lim_ε→ 0+_AZ^H,ε(x)dx.



Denote

    f_A,ε(y)= e^-ε^2|y|^2/2 f_A(y)= e^-ε^2|y|^2/2∏_l=1^31/|y_l|^H+1/α-1_Asin(x,y)dx

and

    Z^H,ε(A)=_^3_Asin(x,y)e^-ε^2|y|^2/2∏_l=1^31/|y_l|^H+1/α-1dx M(dy)
    
    =_^3f_A,ε(y)M(dy).

Let us first prove the possibility to change the order of integration in  (<ref>). To this end, according to Theorem <ref>, it is enough to show that

    _A(_^3|sin(x,y)|^α e^-αε^2|y|^2/2∏_l=1^31/|y_l|^α(H-1)+1dy)^1/αdx<∞.

Estimate

    _^3|sin(x,y)|^α e^-αε^2|y|^2/2∏_l=1^3dy/|y_l|^α(H-1)+1
    ≤_^3(1∧|x||y|)^α e^-αε^2|y|^2/2∏_l=1^3dy/|y_l|^α(H-1)+1:=I.

Through the spherical change of variables 

    y_1=ρsinθcosν; y_2=ρsinθsinν; y_3=ρcosθ; ρ>0, θ∈ [0,π], ν∈ [0,2π],

we get

    I=∫_0^∞∫_0^π∫_0^2π(1∧|x|ρ)^αe^-αε^2ρ^2/2ρ^2(sinθ)^2α(H-1)-1dρ  dθ dφ/ρ^3α(H-1)+3(cosφsinφcosθ)^1-α(H-1)
    ≤ C∫_0^∞(1∧|x|ρ)^α e^-αε^2ρ^2/2ρ^3α(1-H)-1dρ<∞.

Therefore, 

    _A ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H,ε(x)dx=_^3f_A,ε(y)M(dy).

Now 

    _^3|f_A(y)-f_A,ε(y)|^α dy
    ≤_^3|e^-αε^2|y|^2/2-1||f_A(y)|^α dy→ 0, ε→ 0,

in view of the dominated convergence theorem, since ∫_^3|f_A(y)|^α dy<∞ by Theorem <ref>.
Therefore, by Proposition <ref>, 

    _A ∂^3/∂ x_1∂ x_2 ∂ x_3Z^H,ε(x)dx 𝖯⟶Z^H(A), n→∞.



Now we will prove that Z^H(A) is a stochastic measure, that is, that is a σ-additive in probability function of A∈ℬ_f(^3). From the definition and linearity of the integral w.r.t. M it is clear that Z^H(A) is additive. Therefore, it suffices to prove only continuity at zero, which is done in the following proposition.

Let {A_n, n≥ 1}⊂ B_f(^3) be such that for any n ≥ 1, A_n+1⊂ A_n, and ∩_n=1^∞ A_n=∅. Then Z^H(A_n)𝖯⟶ 0, n→∞.
 

Thanks to Proposition <ref>, we need to show that

    _^3|f_A_n(y)|^α dy→ 0, n→∞.

This follows immediately from Proposition <ref> and the continuity of the Lebesgue measure, as

    ||f_A_n||_L^1(^3) + ||f_A_n||_L^2(^3)≤λ(A_n) + λ(A_n)^1/2.

 

{Z^H(A), A ∈ B_f(^3)} is a stochastic measure.


We turn now to integration with respect to the measure Z^H. Since its increments are dependent, the standard integration theory, as described in Section 2, is not available. Nevertheless, we can proceed in a standard way. For a simple function

    g(x)=∑_k=1^na_kI_A_k(x)

with A_k∈ B_f(^3), k=1,…,n, define

    I^H(g)=_^3g(x)Z^H(dx)=∑_k=1^na_kZ^H(A).

Observe that in this case

    I^H(g)=_^3∑_k=1^na_k f_A_k(y)M(dy)=_^3 A^Hg(y)M(dy),

where A^H is given by (<ref>). Then it is natural to define for arbitrary g

    I^H(g)=_^3f(x)Z^H(dx)=_^3A^Hg(y)M(dy).

Thanks to Proposition <ref>,  this is well defined for any g ∈ L^1(^3) ∩ L^2(^3). Moreover, the map I^H is continuous in the following sense.

Let {g_n,n≥ 1}⊂ L^1(^3)∩ L^2(^3) be such that g_n→ g, n→∞, both in L^1(^3) and in L^2(^3). Then

    I^H(g_n)𝖯⟶I^H(g), n→∞.



From linearity of operator A^H and that of integral w.r.t. M we have

    I^H(g_n)-I^H(g)=I^H(g_n-g).

Therefore, it is enough to prove that

    I^H(g_n-g)→ 0, n𝖯⟶∞,

which follows from Propositions <ref> and <ref>.

In particular, the integral I^H(f) = ∫_^3 f(x)Z^H(dx) may be equivalently defined as a limit in probability of integrals of simple functions approximating f in L^1(^3)∩ L^2(^2). Another important observation is that our definition agrees with that given in <cit.>. 

For any f∈ L^1(^3)∩  L^2(^3), the convergence in probability (<ref>) holds.


The proof is the same as in Theorem <ref>.





§ WAVE EQUATION WITH COLOURED STABLE NOISE

Let us return to the wave equation with coloured Sα S noise:

    {(   ∂ ^2/∂ t^2- a^2 Δ)U(x,t)=Ż^H(x), x∈^3, t>0,
       U(x,0)=0, 
       ∂ U/∂ t(x,0)= 0.
    .

This equation was already studied in <cit.>, wherein we proved that its candidate solution given by Kirchhoff's formula

    U(x,t)=1/4π a_y:|x-y|<at1/|x-y|Z^H(dy)

is a weak (generalized) solution, that is for any θ (x,t)∈ C_fin^∞(^3×^+) almost surely it holds that

    ∫_0^∞_^3U(x,t)(∂^2/∂ t^2θ (x,t)-a^2Δθ (x,t))dx dt=∫_0^∞_^3θ (x,t)Z^H(dx)dt.

In <cit.>, the integral in (<ref>) was understood in the sense (<ref>). Nevertheless, thanks to Theorem <ref>, this agrees with the definition (<ref>) taken in the present paper. Indeed, defining

    f_t,x(y)=1/4π a|x-y|I_|x-y|<at

so that

    U(x,t)=_^3f_x,t(y)Z^H(dy),

we have f_t,x∈ L^1(^3)∩ L^2(^3).
As a result, 

    U(x,t)=_^3A^H f_x,t(y) M(dy)
    
    =1/a_^3sin(x,y)∏_l=1^3 y_l/|y_l|^H+1/α-11-cos at|y|/|y|^2M(dy)

(the expression for A_H f_x,t(y) is computed in <cit.>). 




§ SAMPLE PROPERTIES OF SOLUTION TO WAVE EQUATION WITH COLOURED STABLE NOISE


Write the random field U(x,t) as its LePage series: 

    U(x,t)=C_α/a∑_k=1^∞Γ_k^-1/αφ(ξ_k)^-1/α∏_l=1^3ξ_k,l/|ξ_k,l|^H+1/α-1sin(x,ξ_k)1-cos at|ξ_k|/|ξ_k|^2g_k.

For notational simplicity assume that the underlying probability space has the following structure:

    (Ω, ℱ, 𝖯)=(Ω_Γ×Ω_ξ×Ω_g, ℱ_Γ⊗ℱ_ξ⊗ℱ_g, 𝖯_Γ⊗𝖯_ξ⊗𝖯_g),

and for all ω= (ω_Γ,ω_ξ,  ωg), k≥ 1 : Γ_k(ω)=Γ_k(ω_Γ), ξ_k(ω)=ξ_k(ω_ξ), g_k(ω)=g_k(ω_g).

1. The random field U has a modification, which is γ-Hölder continuous in t and locally in X for any γ∈ (0,(3H-1)∧ 1). Moreover,  for any δ>0 this modification satisfies

    sup_|x'|, |x”|≤ R,t',t”∈ [0,T] 
    |x'-x”|≤ h,
    |t'-t”|≤ h|U(x',t')-U(x”,t”)|≤ C(ω)h^(3H-1)∧ 1|log h|^3/α-1/2+δ

for all R>0 and all h>0 small enough.

2. If additionally H ∈(2/3, 1), then this modification is absolutely continuous in each variable.


1. Estimate

    _g[|U(x',t')-U(x”,t”)|^2]
    ≤ 2 (_g[|U(x',t')-U(x”,t')|^2]+_g[|U(x”,t')-U(x”,t”)|^2])

Let h ∈(0,1/2). Define

    a_1(h)=sup_
    t ∈ [0,T], x',x”∈^3 
    |x'-x”|≤ h
    _g[|U(x',t)-U(x”,t)|^2],
    
    a_2(h)=sup_|x|≤ R, t',t”∈ [0,T]
    |t'-t”|≤ h 
    _g[|U(x,t')-U(x,t”)|^2].

Using the LePage representation, estimate

    _g[|U(x',t)-U(x”,t)|^2]
    ≤ a^-2 C_α^2 ∑_k=1^∞Γ_k^-2/αφ(ξ_k)^-2/α|1-cos at|ξ_k||^2/|ξ_k|^4|sin (x',ξ_k) - sin(x”,ξ_k)|^2×
    ×∏_l=1^31/|ξ_k,l|^2H+2/α-2.

From simple inequalities

    |1-cos at|y||/|y|^2≤ 1∧|y|^-2, |sin (x',ξ_k) - sin(x”,ξ_k)|≤ 2∧(|y|· |x'-x”|)

we get the following estimate:

    a_1(h)≤ a^-2 C_α^2 ∑_k=1^∞Γ_k^-2/αφ(ξ_k)^-2/α(1∧|ξ_k|^-4)(4 ∧ (|ξ_k|^2 h^2)  )
    ×∏_l=1^31/|ξ_k,l|^2H+2/α-2
    := a^-2 C_α^2∑_k=1^∞Γ_k^-2/α  Q(h,ξ_k).

Consider

    _ξ[Q(h,ξ_k)]=_^3φ(y)^1-2/α(1∧|y|^-4) (4∧ (|y|^2 h^2))∏_l=1^31/|y_l|^2H+2/α-2dy
    
    =K^3-6/α_^3(1∧|y|^-4) (4∧ (|y|^2 h^2))∏_l=1^3dy/|y_l|^2H-1(|log|y_l||+1)^(1+η)(1-2/α).

Now we make the spherical change of variables (<ref>) and estimate the logarithms as

    (|log|ρ a(ν,θ)||+1)^d≤(|logρ|+ |log|a(ν,θ)||+1)^d
    ≤(|logρ|+1)^d(|log|a(ν,θ)||+1)^d,

where d=(1+η)(2/α-1), and a(ν,θ) is one of the functions sinθcosν, sinθsinν, cosθ. Thus we get

    _ξ[Q(h,ξ_k)]≤ C ∫_0^∞(1∧ρ^-4)(ρ^2h^2∧ 1)(|logρ|+1)^3d/ρ^6H-5
    ×∫_0^2π∫_0^π(|log|sinθcosν||+1)^d(|log|sinθsinν||+1)^d
    
    ×(|log|cosθ||+1)^d|sinθ|dθ  dν  dρ
    ≤ C ∫_0^∞(1∧ρ^-4)(ρ^2h^2∧ 1)(|logρ|+1)^3d/ρ^6H-5dρ
    
    = C(h^2∫_0^1ρ^7-6H(|logρ|+1)^3d dρ
    +∫_1^h^-1h^2ρ^3-6H(|logρ|+1)^3ddρ
    +∫_h^-1^∞ρ^1-6H(|logρ|+1)^3d dρ):=C(I_1+I_2+I_3).

Let us estimate the integrals individually. Obviously, I_1 ≤ C h^2.
Further, for ρ∈[1,h^-1], (|logρ|+1)^3d≤(|log h|+1)^3d, 
whence 

    I_2 ≤ C (|log h|+1)^3d∫_1^1/hρ^3-6H dρ≤ 
    C h^(6H-2)(|log h|+1)^3d.

Finally,

    I_3=h^-1∫_1^∞(ρ/h)^1-6H(logρ/h+1)^3ddρ
    ≤ h^6H-2∫_1^∞ρ^1-6H(logρ+|log h|+1)^3d dρ
    ≤ h^6H-2(|log h|+1)^3d∫_1^∞ρ^1-6H(logρ+1)^3d dρ≤ C h^6H-2(|log h|+1)^3d.

Collecting the estimates, we get that

    _ξ[ Q(h,ξ_k)]≤ Ch^(6H-2)∧ 2(|log h|+1)^3d.

Define b(h)=h^(6H-2)∧ 2|log h|^3d, h ∈(0,1/2). Taking into account (<ref>), (<ref>) and the almost sure convergence of the series ∑_k=1^∞Γ_k^-2/α we get that for any ε>0

    _ξ[∑_n=1^∞a_1(2^-n)/b(2^-n)n^1+ε]<∞

𝖯_Γ-almost surely.
Hence  ∑_n=1^∞a_1(2^-n)/b(2^-n)n^1+ε<∞  𝖯_ξ⊗𝖯_Γ-almost surely. It follows in particular that 

    a_1(2^-n)/b(2^-n)n^1+ε→ 0, n→∞,

𝖯_ξ⊗𝖯_Γ-almost surely. 
Note that, for positive h small enough, b(h) increases and satisfies b(2h)≤ C b(h). Consequently, (<ref>) implies that for almost all (ω_ξ,ω_Γ) ∈Ω_ξ×Ω_Γ

    a_1(h)≤ C(ω_ξ,ω_Γ) h^(6H-2)∧ 2|log h|^3d+1+ε

holds for all sufficiently small h>0.

In order to estimate a_2(h), use the following inequality:

    |(1-cos at'|y|)-(1-cos at”|y|)|≤ 2∧(a|y||t'-t”|).

Then 

    a_2(h)≤ a^-2 C_α^2 ∑_k=1^∞Γ_k^-2/αφ(ξ_k)^-2/α4 ∧(a^2|ξ_k|^2h^2)/|ξ_k|^4
    ×(1∧ (|x|^2|ξ_k|^2))∏_l=1^31/|ξ_k,l|^2H+2/α-2.

Using the same reasoning as above, we get that for almost all (ω_ξ,ω_Γ) ∈Ω_ξ×Ω_Γ

    a_2(h)≤ C (ω_ξ,ω_Γ)h^(6H-2)∧ 2|log h|^3d+1+ε.

As a result, we get for almost all ω_ξ∈Ω_ξ,ω_Γ∈Ω_Γ

    sup_A_R,h_g[|U(x',t')-U(x”,t”)|^2]≤ C(ω_ξ,ω_Γ)h^(6H-2)∧ 2|log h|^3d+1+ε,

for all h small enough, 
where 
    A_R,h = {(x',x”,t',t”)∈^6×[0,T]^2: |x'|, |x”|≤ R,|x'-x”|≤ h,
    |t'-t”|≤ h}.

For a fixed (ω_ξ,ω_Γ) ∈Ω_ξ×Ω_Γ,  U has a centered Gaussian distribution. Therefore (see e.g. <cit.>) there is a modification of U satisfying

    sup_A_R,h|U(x',t')-U(x”,t”)|≤ C(ω)h^(3H-1)∧ 1|log h|^3d/2+1+ε/2.

Recalling that d = (1+η)(2/α -1), we get the required statement by choosing ε and η sufficiently small. 


2. Assuming that the continuous modification of U is chosen, define 

    g(x,y,t)=∂/∂ tA^Hf_x,t=sin(x,y)sin at|y|/|y|^2|y| ∏_l=1^3 y_l/|y_l|^H+1/α-1

and 

    V(x,t)=_^3g(x,y,t)M(dy)=_^3sin(x,y)sin at|y|/|y| ∏_l=1^3 y_l/|y_l|^H+1/α-1M(dy).

With the help of spherical change of variables (<ref>), estimate

    _^3|g(x,y,t)|^α dy≤_^3((1∧|x||y|)(1∧|a||t||y|)/|y| ∏_l=1^3|y_l|^H+1/α-1|y|)^α dy
    
    =∫_0^∞∫_0^π∫_0^2π(1∧|x|ρ)^α(1∧|a||t|ρ)^αρ^2sinθ/ρ^3α H+3-2α|sinθ|^2α H+2-2α
    ×(cosφsinφcosθ)^α-1-α Hdθ  dφ  dρ
    
    =∫_0^∞∫_0^π∫_0^2π(1∧|x|^αρ^α)(1∧|a|^α|t|^αρ^α)
    
    ×(cosφsinφcosθ)^α-1-α H(sinθ)^2α-2α H-1ρ^2α-3α H-1dθ  dφ  dρ
    ≤ C ∫_0^∞(1∧|x|^αρ^α)(1∧|a|^α|t|^αρ^α)ρ^2α-3α H-1dρ<∞;

the integral is convergent since 4α-3α H -1>-1 and  2α-3α H-1<-1 for α∈ (1,2) and  H∈ (2/3,1).
As a result,

    ∫_0^t(_^3|g(x,y,s)|^α dy)^1/αds<∞,

so by Theorem <ref>

    ∫_0^t V(x,s)ds=_^3∫_0^tg(x,y,s)ds  M(dy)=_^3A^H f_x,t(y)M(dy)=U(x,t)

almost surely. Since both sides of the last equality are continuous, and the left-hand side is absolutely continuous, then we get that, almost surely, U is absolutely continuous in t. The absolute continuity in other variables is shown similarly. 




§ DEFINITION AND PROPERTIES OF  A^H


Let g∈ L^1(^3)∩ L^2(^3). Define

    A^Hg(y)=g(y)∏_l=1^3 y_l /|y_l|^H+1/α-1,

where g(y)=_^3e^i(x,y)g(x)dx is the Fourier transform of g.

For any α∈ (0,2), H∈(1/2,1), there exists a constant C(H,α) such that for any g ∈ L^1(^3)∩ L^2(^3), 

    A^Hg_L^α(^3)≤ C(H,α) (g_L^1(^3) + g_L^2(^3)).



We need to estimate _^3|A^Hg(y)|^α dy. Let us split the integral into two: over the unit ball B(0,1) and over its complement B(0,1)^c. To estimate the former, write 

    _B(0,1)|A^Hg(y)|^α dy =_ B(0,1)|g(y)|^α∏_l=1^31/|y_l|^α(H-1)+1dy_l
    ≤g^α_L^1(^3)_ B(0,1)∏_l=1^31/|y_l|^α(H-1)+1dy_l,

and the integral is finite, since H<1. 

To estimate the integral over B(0,1)^c, use the Hölder inequality with p=2/α, q = 2/(2-α) to get

    _B(0,1)^c|A^Hg(y)|^α dy =_ B(0,1)^c|g(y)|^α∏_l=1^31/|y_l|^α(H-1)+1dy_l
    ≤(_ B(0,1)^c|g(y)|^2dy)^α/2(_ B(0,1)^c∏_l=1^3|y_l|^α q(1-H)-qdy_l)^1/q.

In the second integral make the spherical change of variables (<ref>) to obtain

    _B(0,1)^c∏_l=1^3|y_l|^α q(1-H)-qdy_l≤ C ∫_1^∞ρ^3α q(1-H)-3q+2dρ <∞,

since the inequality 3α q(1-H)-3q+2<-1 easily transforms to α (1-2H)<0. For the first integral in (<ref>) we use the Parseval identity: 

    _ B(0,1)^c|g(y)|^2dy ≤_^3|g(y)|^2dy=1/(2π)^3_^3(g(y))^2dy.

This implies the required estimate.



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