Introduction

VMSTA

Modern Stochastics: Theory and Applications

2351-6054 2351-6046

2351-6046

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Mokslininkų g. 2A, 08412 Vilnius, Lithuania

VMSTA168

10.15559/20-VMSTA168

Research Article

Averaging principle for a stochastic cable equation

https://orcid.org/0000-0002-4713-5274

Bodnarchuk

Iryna

ibodnarchuk@knu.ua Taras Shevchenko National University of Kyiv, Volodymyrska Str. 64/13, 01601 Kyiv, Ukraine

2020

21122020

74449467 5102020 22112020 1122020

2020

Open access article under the CC BY license.

We consider the cable equation in the mild form driven by a general stochastic measure. The averaging principle for the equation is established. The rate of convergence is estimated. The regularity of the mild solution is also studied. The orders in time and space variables in the Holder condition for the solution are improved in comparison with previous results in the literature on this topic.

Keywords Averaging principle stochastic cable equation stochastic measure mild solution Hölder regularity

2010 MSC 60G57 60H05 60H15

Alexander von Humboldt Foundation

This research was partially supported by the Alexander von Humboldt Foundation in the framework of the Research Group Linkage Potsdam/Kyiv entitled “Singular diffusions: analytic and stochastic approaches”.

1 Introduction

Averaging methods are important for describing and investigating the asymptotic behavior of dynamical systems. Therefore, the theory of the averaging principle for stochastic differential equations is a fascinating modern topic and many mathematicians work quite actively within this field. For instance, a weak order in averaging for wave equations with L2 -valued Wiener processes is studied in [14]. Bao et al. [3] considered two-time-scale equations with α-stable noises. Strong and weak orders in averaging for stochastic partial differential equations with Wiener processes are given in [10, 11, 13].

The averaging principle for fractional differential equations driven by Lévy noise is established by Shen et al. [31]. Wang and Xu [34] investigated the stochastic averaging method for neutral stochastic delay equations driven by fractional Brownian motion with Hurst parameter H∈(1/2,1) . Other interesting examples of studying of averaging for stochastic differential equations can be found in papers [1, 12, 15, 18].

The averaging principle for equations driven by general stochastic measures is considered in [6, 22, 23, 26, 30].

Our aim here is to establish the averaging principle for the cable equation driven by stochastic measure studied in [25]. For this purpose we also improve the orders of the Hölder condition for the mild solution with respect to space and time variables obtained in [25, Theorem 5.1] (see Theorem 1 below).

Properties of the mild solutions to stochastic partial differential equations are studied in a number of papers. In particular, the existence and uniqueness of a solution for the class of non-autonomous parabolic stochastic partial differential equations defined on a bounded open subset D⊂Rd and driven by an L2(D) -valued fractional Brownian motion with the Hurst index H>1/2 1/2$]]> are proved in [28]. In [2] the ergodic property of the solution to a fractional stochastic heat equation is established. Wave equations with general stochastic measures and α-stable distributions are investigated in the papers [5, 7, 8, 24] and [20, 29, 19], respectively.

Let L0(Ω,F,P) be the set of all real-valued random variables defined on a complete probability space (Ω,F,P) , X be an arbitrary set and B(X) be a σ-algebra of Borel subsets of X . Let μ be a stochastic measure on B(X) , i.e. a σ-additive mapping μ:B(X)→L0(Ω,F,P) . Such μ is also called a general stochastic measure (see, for example, [17, Section 7]). Examples of stochastic measures can be found in [17, 21, 30].

Consider the mild solution to the following equation: (1) ∂uε(t,x)∂t=∂2uε(t,x)∂x2−uε(t,x)+σ(t/ε,x)μ˙(x),uε(0,x)=u0(x),∂uε(t,0)∂x=∂uε(t,L)∂x=0, where (t,x)∈[0,T]×[0,L] , T>0 0$]]>, L>0 0$]]>, ε>0 0$]]>, and μ is a stochastic measure defined on the Borel σ-algebra B([0,L]) .

Let G be the fundamental solution of the homogeneous cable equation, that is (2) G(t,x,y)=e−t4πt ∑n=−∞∞e−(y−x−2nL)24t+e−(y+x−2nL)24t (see, for example, [33, p. 312] or [32, equality (5.69B)]). Then the mild solution of problem (1) is given by the formula (3) uε(t,x)=∫0LG(t,x,y)u0(y)dy+∫[0,L]dμ(y)∫0tG(t−s,x,y)σ(s/ε,y)ds.

We study the convergence uε(t,x)→u¯(t,x),ε→0, where u¯(t,x) is the mild solution of the averaged equation, that is, (4) u¯(t,x)=∫0LG(t,x,y)u0(y)dy+∫[0,L]dμ(y)∫0tG(t−s,x,y)σ¯(y)ds, and (5) σ¯(x)=limt→∞1t∫0tσ(s,x)ds.

The rest of the paper is organized as follows. Section 2 contains some basic facts concerning the estimates of stochastic integrals with respect to general stochastic measures. In Section 3, we study the Hölder regularity of the mild solution of the cable equation with respect to the set of all variables. The averaging principle for the cable equation is established in Section 4.

2 Preliminaries

To prove the convergence of solutions, we will apply an estimate of a stochastic integral using the norm of the Besov space B22α([b,c]) , α∈(1/2,1) , b,c∈R (see, for example, [16]). B22α([b,c]) is the space of functions g:[b,c]→R such that the norm (6) ‖g‖B22α([b,c])=‖g‖L2([b,c])+(∫0c−b(w2,[b,c](g,r))2r−2α−1dr)1/2, is finite. Here w2,[b,c](g,r)=sup0≤h≤r∫bc−h|g(s+h)−g(s)|2ds1/2.

Denote Δkn(L)=((k−1)2−nL,k2−nL],n≥0,1≤k≤2n.

Let Z be an arbitrary set and a function g(y,z):[0,L]×Z→R be such that g(·,z) is continuous on [0,L] for all z∈Z . Put gn(y,z)=g(0,z)1{0}(y)+∑1≤k≤2ng((k−1)2−nL,z)1Δkn(L)(y).

By [27, Lemma 3], the random function η(z)=∫[0,L]g(y,z)dμ(y),z∈Z, has a version η˜(z)=∫[0,L]g0(y,z)dμ(y)+∑n≥1∫[0,L]gn(y,z)dμ(y)−∫[0,L]gn−1(y,z)dμ(y), such that, for all ε>0 0$]]>, ω∈Ω , z∈Z , η˜(z)≤g(0,z)μ([0,L])+∑n≥12nε∑1≤k≤2ng(k2−nL,z)−g((k−1)2−nL,z)212×∑n≥12−nε∑1≤k≤2nμΔkn(L)212. This version is the same for all z∈Z .

According to [16, Theorem 1.2], [21, Lemma 3.2] and [9, inequality (6)], we have (7) η˜(z)≤|g(0,z)μ([0,L])|+C‖g(·,z)‖B22α([0,L])∑n≥12−nε∑1≤k≤2nμΔkn(L)212, where α=ε/2+1/2 , the constant C depends on α,t and does not depend on z,ω .

Here and in what follows the same symbol C denotes some positive constants that may be different in different places of the paper. The precise values of these constants are not important for our purposes.

3 Regularity of the mild solution of a cable equation

Regularity of the mild solution (8) u(t,x)=∫0LG(t,x,y)u0(y)dy+∫[0,L]dμ(y)∫0tG(t−s,x,y)σ(s,y)ds, of a cable equation driven by a general stochastic measure is studied in [25]. It was proved there that the paths of the solution are Hölder continuous. The following conditions was considered.

Condition 1. The function σ(s,y):[0,T]×[0,L]→R has the derivative ∂2σ∂t∂x , which is continuous with respect to the pair of arguments.

Condition 2. The function u0(y)=u0(y,ω):[0,L]×Ω→R is measurable and has the derivative ∂u0∂y , which is continuous with respect to y and bounded for all fixed ω∈Ω .

By [25, Theorem 5.1], if Conditions 1–2 hold, then for all fixed δ>0 0$]]>, γ2<1/18 , and γ1<1/6 , function (8) has a version u˜(t,x) such that (9) |u˜(t1,x1)−u˜(t2,x2)|≤Lu˜(ω)|t1−t2|γ2+|x1−x2|γ1, for all t1,t2∈[δ,T] , x1,x2∈[0,L] and for some Lu˜(ω)>0 0$]]>.

The proof of Theorem 5.1 ([25]) uses the Hölder regularity of the mild solution of a heat equation that is established in [21]. Restrictions γ1<1/6 and γ2<1/18 for relation (9) are due to the restrictions for the corresponding orders of the Hölder condition for the stochastic integrals from the heat equation (see [21, Lemma 5.1 and Lemma 6.1]). But now we can use the results of [4], which improve Hölder continuity orders obtained in [21]. Namely, the Hölder regularity with γ1<1/2 and γ2<1/4 for the mild solution of a parabolic equation with a stochastic measure is proved in [4, Lemma 1 and Lemma 2].

The following assumptions are used in the rest of the paper.

A1. The function u0(y)=u0(y,ω):[0,L]×Ω→R is measurable and Hölder continuous, that is |u0(y1)−u0(y2)|≤Lu0(ω)|y1−y2|β(u0),β(u0)>0. 0.\]]]>

A2. σ(s,y):R+×[0,L]→R is measurable, bounded and Hölder continuous in s∈R+ , y∈[0,L] , that is |σ(s,y)|≤Cσ,|σ(s1,y1)−σ(s2,y2)|≤Lσ(|s1−s2|β(σ)+|y1−y2|β(σ)),1/2<β(σ)<1.

A3. Limit (5) exists and Σ(r,y)=∫0t[σ(s,y)−σ¯(y)]ds is bounded for all t∈R+ , y∈[0,L] . Theorem 1.

Let Assumptions A1–A2 hold. Then there exists a version u˜(t,x) of the random function u(t,x) defined by (8) such that for all fixed δ>0 0$]]>, γ˜2<1/4∧(β(σ)−1/2) and γ˜1<1−1/(2β(σ)) , γ˜1≤β(u0) , we have (10) |u˜(t1,x1)−u˜(t2,x2)|≤Cu˜(ω)|t1−t2|γ˜2+|x1−x2|γ˜1, for all t1,t2∈[δ,T] and x1,x2∈[0,L] and for some constant Cu˜(ω)>0 0$]]>.

Proof.

The reasoning is the same as that used in [25] with some differences. We have divided the proof into 3 steps: the Hölder continuity of the stochastic integral with respect to the space variable; the Hölder continuity of the stochastic integral with respect to the time variable and the Hölder continuity of the function u(t,x) .

Step 1. By obtaining the Hölder condition for the stochastic integral with respect to the space variable we get the analogues of [25, inequalities (3.1) and (3.2)]. We use Assumption A2 instead of Condition 1.

Consider n≠0,1 . By A2, we have F2≤Cσπ(4(t−s))3/2|x1−x2|2(|n|+1)Le−min(y−x1−2n)2,(y−x2−2n)2L24(t−s)≤C(t−s)3/2|x1−x2|·|n|e−(2|n|−1)2L24(t−s). For details see the method used for obtaining bound (35) below. The same estimation holds for term F1 . Therefore, (11) F≤C(t−s)3/2|x1−x2|·|n|e−(2|n|−1)2L24(t−s).

On the other hand, F≤|fn(s,x1,y+h)−fn(s,x1,y)|+|fn(s,x2,y+h)−fn(s,x2,y)|=F˜1+F˜2. Since the function σ is bounded and Hölder continuous (by Assumption A2), and (t−s)−1/2≤T(t−s)−3/2 , we obtain F˜1≤14π(t−s)e−(y−x1−2nL)24(t−s)|σ(s,y+h)−σ(s,y)|+14π(t−s)e−(y+h−x1−2nL)24(t−s)−e−(y−x1−2nL)24(t−s)|σ(s,y+h)|≤C(t−s)3/2Lσhβ(σ)+Cσh2(|n|+1)Le−min(y−x1−2n)2,(y+h−x1−2n)2L24(t−s)≤C(t−s)3/2hβ(σ)·|n|e−(2|n|−1)2L24(t−s). The same estimate holds for term F˜2 . Hence, (12) F≤C(t−s)3/2hβ(σ)·|n|e−(2|n|−1)2L24(t−s).

Raise the inequality (12) to the power θ1 and multiply by inequality (11) raised to the power 1−θ1 , for an arbitrary θ1∈(0,1) . We have (13) F≤C(t−s)3/2|x1−x2|1−θ1hθ1β(σ)·|n|e−(2|n|−1)2L24(t−s), for all n≠0 .

For the same reason, (14) |fn(s,−x2,y+h)−fn(s,−x1,y+h)−fn(s,−x2,y)+fn(s,−x1,y)|≤C(t−s)3/2|x1−x2|1−θ1hθ1β(σ)·|n|e−min(2n)2,(2n−2)2L24(t−s), for all n≠0,1 .

Further, we can repeat the proof of Theorem 3.1 ([25]). Instead of using bounds (3.1) and (3.2) of [25], we can use inequalities (13) and (14) above. The integral from (6) is finite if θ1β(σ)>1/2 1/2$]]>. Thus we get the Hölder condition with respect to the space variable of the order γ˜1=1−θ1<1−12β(σ)<12.

Besides, as mentioned above, we use [4, Lemma 1 and Lemma 2] instead of [21, Lemma 5.1 and Lemma 6.1] in the case of n=0,1 .

Step 2. The reasoning is the same in the case of the Hölder property with respect to the time variable. Given x and t1<t2 use the notation of paper [25], that is f¯n(t,s,y)=e−(t−s)4π(t−s)e−(y−x−2nL)24(t−s)+e−(y+x−2nL)24(t−s)σ(s,y),n∈Z,G¯(1)(t,s,y)=∑n≠0,1f¯n(t,s,y), and (15) g¯(y)=∫0t2G¯(1)(t2,s,y)ds−∫0t1G¯(1)(t1,s,y)ds.

Moreover, denote G¯1(t,s,y)=∑n≠0,1e−(t−s)4π(t−s)e−(y−x−2nL)24(t−s)+e−(y+x−2nL)24(t−s).

Now we consider the case of n≠0,1 , since the case of n=0,1 is the same as in the [25] with reference to [4, Lemma 1 and Lemma 2] instead of [21, Lemma 5.1 and Lemma 6.1] respectively. Namely, we get Hölder continuity of the order 1/4 for these terms.

We use the change of variables s→s+t2−t1 in the first integral of (15) and obtain g¯(y)=∫t1−t2t1G¯1(t1,s,y)σ(s+t2−t1,y)ds−∫0t1G¯1(t1,s,y)σ(s,y)ds=∫t1−t20G¯(1)(t2,s+t2−t1,y)σ(s+t2−t1,y)ds+∫0t1G¯1(t1,s,y)σ(s+t2−t1,y)−σ(s,y)ds=g¯1(y)+g¯2(y).

Therefore, (17) |g¯1(y+h)−g¯1(y)|≤Chβ(σ)∫t1−t20e−L24(t1−s)(t1−s)3/2ds≤Chβ(σ)|t2−t1|, where we used the estimate e−L24(t1−s)(t1−s)3/2≤1t13/2≤1δ3/2≤C, for s in the domain of integration.

Then, by assumption A2, we get (18) |g¯2(y)|≤∫0t1G¯1(t1,s,y)σ(s+t2−t1,y)−σ(s,y)ds≤C|t2−t1|β(σ).

On the other hand, the same reasoning as that used in obtaining bound (12) proves (19) |g¯2(y+h)−g¯2(y)|≤Chβ(σ).

Now we raise the inequality (19) to the power θ2 and multiply by inequality (18) raised to the power 1−θ2 , for an arbitrary θ2∈(0,1) . Thus we see that (20) |g¯(y+h)−g¯(y)|≤C|t2−t1|1−θ2hθ2β(σ).

Consequently, taking to consideration (17), (21) w2(g¯,r)≤C|t2−t1|(1−θ2)β(σ)rθ2β(σ).

From this point we can repeat the proof of Theorem 4.1 ([25]). Instead of using (4.2) of [25], we use bounds (20) and (21) above. Since the integral from (6) is finite for θ2β(σ)>1/2 1/2$]]>, 0<1−θ2<1−12β(σ)<12.

Therefore, we obtain the Hölder condition with respect to the time variable of the order γ˜2=minβ(σ)−12,14, where we also count the case n=0,1 .

Step 3. The last part of the proof is analogous to the proof of Theorem 5.1 in [25]. The integral ∫0LG(t,x,y)u0(y)dy satisfies condition (10). To prove the Hölder regularity with respect to the space variable we consider x1<x2 and denote G±(t,x,y)=e−t4πt ∑n=−∞∞e−(y±x−2nL)24t.

To prove the Hölder regularity with respect to the time variable we consider δ≤t1<t2≤T . We get |G+(t1,x,y)−G+(t2,x,y)|=|e−t14πt1 ∑n=−∞∞e−(y+x−2nL)24t1+e−t24πt2 ∑n=−∞∞e−(y+x−2nL)24t2|≤e−t1−e−t24πt1 ∑n=−∞∞e−(y+x−2nL)24t1+e−t24πt1 ∑n=−∞∞|e−(y+x−2nL)24t1−e−(y+x−2nL)24t2|+e−t214πt1−14πt2 ∑n=−∞∞e−(y+x−2nL)24t2=J1+J2+J3.

According to estimates (34), (25) and (27) from Section 4, J1≤|t1−t2|4πt1 ∑n=−∞∞e−(y+x−2nL)24t1≤|t1−t2|4πδ∑n≠0e−min(2n)2,(2n−2)2L24t1+1≤C|t1−t2|.

Finally, J3≤C4πt1t2t2−t1 ∑n=−∞∞e−(y+x−2nL)24t2≤Cδ4πt2−t1≤Ct2−t1.

In consequence, |G+(t1,x,y)−G+(t2,x,y)|≤Ct2−t1.

Likewise, we get the same estimate for |G+(t1,x,y)−G+(t2,x,y)| . Hence, ∫0LG(t1,x,y)u0(y)dy−∫0LG(t2,x,y)u0(y)dy≤C(ω)t2−t1.

The rest of the proof runs as respective part of the proof of Theorem 5.1 in [25] with the use of Steps 1 and 2 instead of [25, Theorems 3.1 and 4.1]. □

4 Averaging principle

In this section we consider the random functions uε and u¯ given by equations (3) and (4).

Theorem 2.

Assume that Assumptions A1–A3 hold. Then there exist versions of uε and u¯ such that for any γ<12(1−12β(σ)) , we have supε∈(0,T],t∈[0,T],x∈[0,L]ε−γ|uε(t,x)−u¯(t,x)|<+∞a. s.

Proof.

By Theorem 1, if Assumptions A1–A2 hold, functions (3) and (4) have continuous in (t,x) versions. We consider these versions.

Let ε>0 0$]]>, t∈[0,T] , x∈[0,L] be fixed. Put (22) g(t,x,y)=∫0tG(t−s,x,y)[σ(s/ε,y)−σ¯(y)]ds, where the fundamental solution G(t,x,y) is given by (2).

Denote Dn±=(y±x−2nL)24 and Σε(r)=∫0r[σ(t/ε−τ,y)−σ¯(y)]dτ,0≤r≤t/ε. Then |Σε(r)|≤CΣ , where CΣ does not depend on ε, while ∫0t[σ(s,y)−σ¯(y)]ds is bounded by A3.

In the integral from (22) we substitute τ=(t−s)/ε and obtain |g(t,x,y)|=ε∫0t/εG(τε,x,y)[σ(t/ε−τ,y)−σ¯(y)]dτ=ε|∫0t/εe−τε4πτ ∑n=−∞∞e−(y−x−2nL)24τε+e−(y+x−2nL)24τε×[σ(t/ε−τ,y)−σ¯(y)]dτ|≤ε4π ∑n=−∞∞∫0t/εe−Dn−τετ[σ(t/ε−τ,y)−σ¯(y)]dτ+ ∑n=−∞∞∫0t/εe−Dn+τετ[σ(t/ε−τ,y)−σ¯(y)]dτ.

We have (23) ∑n=−∞∞∫01e−Dn±τετ[σ(t/ε−τ,y)−σ¯(y)]dτ≤2Cσ∫01 ∑n=−∞∞e−Dn±τετdτ≤4Cσsupτ≤1 ∑n=−∞∞e−Dn±τε=4Cσ ∑n=−∞∞e−Dn±ε and (24) | ∑n=−∞∞∫1t/εe−Dn±τετ[σ(t/ε−τ,y)−σ¯(y)]dτ= ∑n=−∞∞∫1t/εe−Dn±τετdΣε(τ)= ∑n=−∞∞e−Dn±τετΣε(τ)|1t/ε− ∑n=−∞∞∫1t/εΣε(τ)e−Dn±τεDn±ετ5−12τ3dτ≤2CΣsup1≤τ≤t/ε ∑n=−∞∞e−Dn±τε+CΣ ∑n=−∞∞∫1t/εDn±e−Dn±τεετ5dτ≤2CΣ ∑n=−∞∞e−Dn±T+CΣ∫1t/ε∑n=−∞∞Dn±e−Dn±τεετ5dτ.

Now we estimate e−Dn±s,s∈[0,T] . For n=0 , e−Dn±s=e−(y±x)24s≤1. Consider n≠0 . In this case (25) e−Dn−s≤e−(2|n|−1)2L24sande−Dn+s≤e−min(2n)2,(2n−2)2L24s.

Since (26) ∑n≥0e−n2a≤∑n≥0e−na=(1−e−a)−1≤(1−e−b)−1,a≥b>0, 0,\]]]> and ∑n≥1e−n2a≤e−a(1−e−a)−1≤e−b(1−e−b)−1,a≥b>0, 0,\]]]> we conclude that, for s∈[0,T] , ∑n≠0,n=−∞∞e−Dn−s≤ ∑n≠0,n=−∞∞e−(2|n|−1)2L24s≤2e−L24s∑n≥0e−n2L2s≤2e−L24s1−e−L2s−1≤21−e−L2T−1.

Analogously, (27) ∑n≠0,n=−∞∞e−Dn+s≤ ∑n≠0,n=−∞∞e−min(2n)2,(2n−2)2L24s=∑n≥1e−(2n)2L24s+∑n≥1e−(2n−2)2L24s≤21−e−L2T−1.

Substitute obtained bounds to (23) and (24). We thus get ∑n=−∞∞∫01e−Dn±τετ[σ(t/ε−τ,y)−σ¯(y)]dτ≤4Cσ1+21−e−L2T−1 and ∑n=−∞∞∫1t/εe−Dn±τετ[σ(t/ε−τ,y)−σ¯(y)]dτ≤2CΣ1+21−e−L2T−1+CΣ∫1t/ε∑n=−∞∞Dn±e−Dn±τεετ5dτ.

Hence, |g(t,x,y)|≤Cε[C+|∫1t/ε∑n=−∞∞Dn−e−Dn−τεετ5dτ|+|∫1t/ε∑n=−∞∞Dn+e−Dn+τεετ5dτ|].

Next we estimate the integrals from the last relation. For |y|≤L and |x|≤L it is clear that (28) Dn±≤(1+|n|)2L2.

Therefore, ∑n=−∞∞Dn−e−Dn−τε=D0−e−D0−τε+ ∑n≠0,n=−∞∞Dn−e−Dn−τε≤D0−e−D0−τε+2L2e−L24τε∑n≥0(n+2)2e−nL2τε≤D0−e−D0−τε+CL2e−L24τε(1−e−L2T)−1. where we use the relations ∑n≥1n2e−na=(e−a+e−2a)(1−e−a)−3≤2e−a(1−e−a)−3≤2(1−e−a)−1,∑n≥1ne−na≤∑n≥1n2e−na≤2(1−e−a)−1,a>0, 0\hspace{0.1667em},\end{aligned}\]]]> and estimate (26). Then, due to the fact that maxD>0De−Dτε=τεe−1 0}{\max }D{e^{-\frac{D}{\tau \varepsilon }}}=\tau \varepsilon {e^{-1}}$]]>, we have (29) ∑n=−∞∞Dn−e−Dn−τε≤Cτε,C=C(L,T).

Consequently, ∫1t/ε∑n=−∞∞Dn−e−Dn−τεετ5dτ≤C∫1t/ε1τ3dτ≤2C.

In the same way we obtain (30) ∑n=−∞∞Dn+e−Dn+τε=D0+e−D0+τε+D1+e−D1+τε+ ∑n≠0,1,n=−∞∞Dn+e−Dn+τε≤2τεe−1+L2∑n≥1(n+1)2e−(2n)2L24τε+L2∑n≥2(n+1)2e−(2n−2)2L24τε≤Cτε(1−e−L2T)−1≤Cτε, which yields ∫1t/ε∑n=−∞∞Dn+e−Dn+τεετ5dτ≤2C.

Therefore, we get |g(t,x,y)|≤C(σ,L,T)ε=Cε and so (31) |g(t,x,y+h)−g(t,x,y)|≤Cε,‖g(t,x,·)‖L2[0,L]≤Cε,w2,[0,L]2(g,r)≤Cε.

By condition A3 and finiteness of ∑n=−∞∞e−Dn±s , we obtain (33) |I2|≤C2Lσhβ(σ)∫0te−(t−s)4π(t−s)ds≤C2Lσhβ(σ)t≤Chβ(σ),∫0L−hI22dy≤Ch2β(σ).

Denote Dn±h=(y+h±x−2nL)24. Recall that here h∈[0,L] and we consider the integrals in (32) such that y+h≤L . So |I1|=∫0te−(t−s)4π(t−s) ∑n=−∞∞e−Dn−ht−s−e−Dn−t−s[σ(s/ε,y+h)−σ¯(y+h)]ds+∫0te−(t−s)4π(t−s) ∑n=−∞∞e−Dn+ht−s−e−Dn+t−s[σ(s/ε,y+h)−σ¯(y+h)]ds=|I1−+I1+|.

Since e−(t−s)≤1 , for t−s≥0 , we conclude by Assumption A2 that |I1−|≤2Cσ∫0t14π(t−s) ∑n=−∞∞e−Dn−ht−s−e−Dn−t−sds=2Cσ|∑n=0,±1∫0t14π(t−s)e−Dn−ht−s−e−Dn−t−sds+∫0t14π(t−s)∑|n|≥2e−Dn−ht−s−e−Dn−t−sds|.

For n=0,±1 , the values of Dn−h and Dn− are bounded by 4L2 . Thus, the same reasoning as that used in obtaining the bound of I2 in [22, inequality (13)] proves ∫0t14π(t−s)e−Dn−ht−s−e−Dn−t−sds≤C|hlnh|≤Chβ(σ).

Consider the second integral. We have ∑|n|≥2e−Dn−ht−s−e−Dn−t−s≤∑|n|≥22h|y+h2−x−2nL|4(t−s)e−min{Dn−h,Dn−}t−s, where we use the following relation (34) |e−a−e−b|≤e−min{a,b}|a−b|,a,b≥0.

According to (28), |y+h2−x−2nL|≤2(1+|n|)2L, and analogously to (25) we obtain e−min{Dn−h,Dn−}t−s≤e−(2|n|−1)2L24(t−s),|n|≥2.

Hence, (35) ∫0t14π(t−s)∑|n|≥2e−Dn−ht−s−e−Dn−t−sds≤Ch∫0t14(t−s)3/2∑n≥2(1+n)2L2e−(2n−1)2L24(t−s)ds.

Then we use the same reasoning as in getting (29) and have ∑n≥2(1+n)2L2e−(2n−1)2L24(t−s)≤C4(t−s).

It follows that ∫0t14π(t−s)∑|n|≥2e−Dn−ht−s−e−Dn−t−sds≤Cht≤Ch, and finally that |I1−|≤Chβ(σ).

The same estimate holds for term |I1+|≤Chβ(σ) . Hence, ∫0L−hI12dy≤Ch2β(σ).

Taking into account (33), we deduce that ∫0L−h|g(t,x,y+h)−g(t,x,y)|2dy≤Ch2β(σ),w2,[0,L]2(g,r)≤Cr2β(σ).

Raise the latter inequality to the power θ and multiply by inequality (31) raised to the power 1−θ , for an arbitrary θ∈(0,1) . We have w2,[0,L]2(g,r)≤Cr2θβ(σ)ε1−θ.

Thus, for any γ<12(1−12β(σ)) , there exists α<θβ(σ) such that ‖g‖B22α([b,c])≤Cε+Cε(1−θ)/2(∫0tr2θβ(σ)−2α−1dr)12≤Cεγ.

Consequently, by relation (7), |uε(t,x)−u¯(t,x)|=∫[0,L]g(t,x,y)dμ(y)≤|g(0,z)μ([0,L])|+C‖g(·,z)‖B22α([0,L])∑n≥12−nε∑1≤k≤2nμΔkn(L)212≤Cεγ|μ([0,L])|+∑n≥12−nε∑1≤k≤2nμΔkn(L)212≤C(ω)εγ, where the sum with the stochastic measure is finite in view of [21, Lemma 3.1], and the constant C(ω) does not depend on t,x and ε. □

Acknowledgement

The author is grateful to the anonymous referee for careful reading of the paper and valuable comments and suggestions.

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