1 Introduction
In this paper, we consider the stochastic equation, which can formally be written as
where $(t,x)\in [0,\hspace{2.5pt}T]\times \mathbb{R}$, and μ is a stochastic measure defined on $\mathcal{B}(\mathbb{R})$ (Borel σ-algebra in $\mathbb{R}$). For μ we assume only σ-additivity in probability, assumptions for f, g, σ and ${u_{0}}$ are given in Section 3. We consider the solution to the formal equation (1) in the mild form (see (5) below).
(1)
\[ \frac{\partial u}{\partial t}=\frac{{\partial ^{2}}u}{\partial {x^{2}}}+f(t,x,u(t,x))+\frac{\partial g}{\partial x}(t,x,u(t,x))+\sigma (t,x)\frac{\partial \mu }{\partial x},\hspace{1em}u(0,x)={u_{0}}(x),\]If $f=0$ and $g(x,t,v)={v^{2}}/2$ then (1) is the Burgers equation, for $g=\mathrm{const}$ we get the heat equation.
In the paper, we prove the existence and uniqueness of the solution and obtain its ${\mathsf{L}^{2}}$-continuity. In addition, we obtain that the averaging principle holds for equation (1).
Equation (1) driven by the Wiener process was studied in [7] for $x\in \mathbb{R}$, in [6] and [8] for $x\in [0,1]$. Equation driven by Lévy process was considered in [9]. The main reason to study equation (1) is that it is a generalization of the Burgers equation which is very important in fluid mechanics. The stochastic Burgers equation was studied, for example, in [4], [11], [14], [16, Chapter 18], [26], [27]. All these equations are driven by Gaussian processes or Lévy processes. In our paper, we consider a more general integrator. Our proofs are based on methods and results of [7].
Stochastic equations driven by stochastic measures were studied, for example, in [1] (general parabolic equation), [2] (wave equation), [17] (heat equation). The averaging principle for such equations was considered in [3], [18], [13]. The detailed theory of stochastic measures is presented in [20]. Note that in all these publications, the functions in the equations are assumed to be bounded.
The rest of the paper is organized as follows. In Section 2 we have compiled some basic facts about stochastic measures. The precise formulation of the problem and our assumptions are given in Section 3, some regularity properties of the stochastic integral are studied in Section 4, and one auxiliary equation is considered in Section 5. The existence and uniqueness of the solution to equation (1) are proved in Section 6. The averaging principle for our equation is obtained in Section 7.
2 Preliminaries
In this section, we give basic information concerning stochastic measures in a general setting. In equation (1), μ is defined on Borel subsets of $\mathbb{R}$.
Let ${\mathsf{L}_{0}}={\mathsf{L}_{0}}(\Omega ,\mathcal{F},\mathsf{P})$ be the set of all real-valued random variables defined on the complete probability space $(\Omega ,\mathcal{F},\mathsf{P})$. Convergence in ${\mathsf{L}_{0}}$ means the convergence in probability. Let $\mathsf{X}$ be an arbitrary set and $\mathcal{B}$ be a σ-algebra of subsets of $\mathsf{X}$.
Definition 1.
A σ-additive mapping $\mu :\hspace{2.5pt}\mathcal{B}\to {\mathsf{L}_{0}}$ is called stochastic measure (SM).
We do not assume the moment existence or martingale properties for SM. In other words, μ is ${\mathsf{L}_{0}}$-valued measure.
Important examples of SMs are orthogonal stochastic measures, α-stable random measures defined on a σ-algebra for $\alpha \in (0,1)\cup (1,2]$ (see [21, Chapter 3]). Theorem 8.3.1 of [10] states the conditions under which the increments of a real-valued process with independent increments generate an SM.
Many examples of the SMs on the Borel subsets of $[0,T]$ may be given by the Wiener-type integral
We note the following cases of processes ${X_{t}}$ in (2) that generate SM.
The following analogue of the Nikodym theorem is satisfied for SMs.
Theorem 1 ([5, Theorem 8.6]).
Applying Theorem 1, we can construct the following example of the SM on all Borel subsets of $\mathbb{R}$:
where ${W_{t}}$ is the Wiener process, $\xi (t)$ is an adapted process such that ${\textstyle\int _{\mathbb{R}}}\mathsf{E}\xi {(t)^{2}}\hspace{0.1667em}dt<\infty $, the limit is taken in ${\mathsf{L}^{2}}(\Omega )$.
(3)
\[ \mu (A)={\int _{\mathbb{R}}}{\mathbf{1}_{A}}(t)\xi (t)\hspace{0.1667em}d{W_{t}}:=\underset{T\to \infty }{\lim }{\int _{[-T,T]}}{\mathbf{1}_{A}}(t)\xi (t)\hspace{0.1667em}d{W_{t}},\]For deterministic measurable functions $g:\mathsf{X}\to \mathbb{R}$, an integral of the form ${\textstyle\int _{\mathsf{X}}}g\hspace{0.1667em}d\mu $ is studied in [20, Chapter 1] (see also [10, Chapter 7]). In particular, every bounded measurable g is integrable w.r.t. any SM μ.
To estimate the stochastic integral, we will use the following statement.
Lemma 1 (Lemma 3.1 [17], Lemma 2.3 [20]).
Let ${f_{l}}:\hspace{2.5pt}\mathsf{X}\to \mathbb{R}$, $l\ge 1$, be measurable functions such that $\bar{f}(x)={\textstyle\sum _{l=1}^{\infty }}|{f_{l}}(x)|$ is integrable w.r.t. μ. Then
We consider the Besov spaces ${B_{22}^{\alpha }}([c,d])$. Recall that the norm in this classical space for $0<\alpha <1$ may be introduced by
where
By C and $C(\omega )$ we will denote positive constants and positive random constants, respectively, whose values may change.
3 The problem
We consider (1) in the mild form
where equality holds a.s. for each $t\hspace{-0.1667em}\in \hspace{-0.1667em}[0,T]$ for almost all $x\hspace{-0.1667em}\in \hspace{-0.1667em}\mathbb{R}$, $u\hspace{-0.1667em}\in \hspace{-0.1667em}\mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$. Here
is the heat kernel, and μ is an SM on $\mathcal{B}(\mathbb{R})$.
(5)
\[ \begin{aligned}{}u(t,x)& ={\int _{\mathbb{R}}}p(t,x-y){u_{0}}(y)\hspace{0.1667em}dy\\ {} & \hspace{1em}+{\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)f(s,y,u(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}-{\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial p}{\partial y}(t-s,x-y)g(s,y,u(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}+{\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y),\end{aligned}\]We will refer to the following assumptions on elements of (5).
Assumption A1.
${u_{0}}(y)={u_{0}}(y,\omega ):\mathbb{R}\times \Omega \to \mathbb{R}$ is measurable and ${u_{0}}(y)={u_{0}}(y,\omega )\in {\mathsf{L}^{2}}(\mathbb{R})$ for each fixed ω.
Assumption A2.
$f(s,y,r):[0,T]\times {\mathbb{R}^{2}}\to \mathbb{R}$ is a Borel function satisfying the linear growth and Lipschitz-type condition:
\[\begin{aligned}{}|f(s,y,r)|& \le {a_{1}}(y)+K|r|,\\ {} |f(s,y,{r_{1}})-f(s,y,{r_{2}})|& \le ({a_{2}}(y)+L|{r_{1}}|+L|{r_{2}}|)|{r_{1}}-{r_{2}}|\end{aligned}\]
for all $s\in [0,T]$, $y,{r_{i}}\in \mathbb{R}$ and for some constants K, L and nonnegative functions ${a_{i}}\in {\mathsf{L}^{2}}(\mathbb{R})$.
Assumption A3.
The function g is of the form
where
\[ {g_{1}}(s,y,r):[0,T]\times {\mathbb{R}^{2}}\to \mathbb{R},\hspace{1em}{g_{2}}(s,r):[0,T]\times \mathbb{R}\to \mathbb{R}\]
are Borel functions satisfying the following linear and quadratic growth conditions:
\[\begin{aligned}{}|{g_{1}}(s,y,r)|& \le {b_{1}}(y)+{b_{2}}(y)|r|,\\ {} |{g_{2}}(s,r)|& \le K|r{|^{2}},\end{aligned}\]
for all $s\in [0,T]$, $y,r\in \mathbb{R}$, where K is a constant, and
\[ {b_{1}}\in {\mathsf{L}^{1}}(\mathbb{R})\cap {\mathsf{L}^{2}}(\mathbb{R}),\hspace{1em}{b_{2}}\in {\mathsf{L}^{2}}(\mathbb{R})\cap {\mathsf{L}^{\infty }}(\mathbb{R})\]
are nonnegative functions. Moreover, g satisfies the following Lipschitz-type condition:
for all $s\in [0,T]$, $y,{r_{i}}\in \mathbb{R}$ and for some constant L and nonnegative function ${b_{3}}\in {\mathsf{L}^{2}}(\mathbb{R})$.
If A4 and A5 hold, then, by Theorem of [1] (or Theorem 3.1 [20]), the random function
\[ \vartheta (t,x)={\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y)\]
has a version $\widetilde{\vartheta }(t,x)$ such that for any fixed
\[ \delta \in (0,T),\hspace{1em}M > 0,\hspace{1em}0 < {\gamma _{1}} < 1/2,\hspace{1em}0 < {\gamma _{2}} < 1/4\]
there exists ${L_{\widetilde{\vartheta }}}(\omega )$ such that
(6)
\[ \begin{array}{r}\displaystyle |\widetilde{\vartheta }({t_{1}},{x_{1}})-\widetilde{\vartheta }({t_{2}},{x_{2}})|\le {L_{\widetilde{\vartheta }}}(\omega )(|{t_{1}}-{t_{2}}{|^{{\gamma _{2}}}}+|{x_{1}}-{x_{2}}{|^{{\gamma _{1}}}}),\\ {} \displaystyle {t_{i}}\in [\delta ,T],\hspace{1em}|{x_{i}}|\le M,\hspace{1em}i=1,2.\end{array}\]4 Regularity property of the stochastic integral
In this section, we obtain some properties of $\widetilde{\vartheta }$.
For any $j\in \mathbb{R}$ and all $n\ge 0$, put
The following estimate is a key tool for the estimates of the stochastic integral.
Lemma 2 (Lemma 3.2 [17]).
Let μ be defined on $\mathcal{B}(\mathbb{R})$, Z be an arbitrary set, and $q(z,x):Z\times [j,j+1]\to \mathbb{R}$ be a function such that for some $1/2<\alpha <1$ and for each $z\in Z$ $q(z,\cdot )\in {B_{22}^{\alpha }}\left([j,j+1]\right)$. Then the random function
has a version $\tilde{\eta }(z)$ such that for some constant C (independent of z, j, ω) and each $\omega \in \Omega $,
Further, we study the properties of stochastic integral ϑ.
Proof.
To prove the continuity in ${\mathsf{L}^{2}}(\mathbb{R})$, for fixed $\omega \in \Omega $ and arbitrary ${t_{n}}\to {t_{0}}$, consider
\[ \| \widetilde{\vartheta }({t_{n}})-\widetilde{\vartheta }({t_{0}}){\| _{{\mathsf{L}_{2}}(\mathbb{R})}^{2}}={\int _{\mathbb{R}}}|\widetilde{\vartheta }({t_{n}},x)-\widetilde{\vartheta }({t_{0}},x){|^{2}}\hspace{0.1667em}dx.\]
By (6), $\widetilde{\vartheta }$ is continuous in both variables, therefore $\widetilde{\vartheta }({t_{n}},x)\to \widetilde{\vartheta }({t_{0}},x)$ for each x.We will find $g\in {\mathsf{L}^{2}}(\mathbb{R})$ such that
then the dominated convergence theorem implies our statement.
We apply Lemma 2 for
Our considerations will imply that the function $q(z,y)$ is continuous in y (see (14) and (15) below), thus the conditions of Lemma 2 are satisfied.
We fix $\omega \in \Omega $, $\theta >1$ and estimate
(10)
\[ \begin{aligned}{}|\widetilde{\vartheta }(t,x){|^{2}}& ={\Big|{\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y)\Big|^{2}}\\ {} & ={\Big|\sum \limits_{j\in \mathbb{Z}}{\int _{(j,j+1]}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y)\Big|^{2}}\\ {} & \le \sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{-\theta }}\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}{\Big|{\int _{(j,j+1]}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y)\Big|^{2}}\\ {} & =C\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}{\Big|{\int _{(j,j+1]}}q(t,x,y)\hspace{0.1667em}d\mu (y)\Big|^{2}}\\ {} & \stackrel{\text{(8)}}{\le }C\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}\Big(|q(t,x,j)\mu ((j,j+1]){|^{2}}\\ {} & \hspace{1em}+\| q(t){\| _{{B_{\alpha }^{2,2}}([j,j+1])}^{2}}\Big\{\sum \limits_{n\ge 1}{2^{-n(2\alpha -1)}}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}\Big).\end{aligned}\]Below we will use the following simple estimates:
(11)
\[ \begin{aligned}{}{\int _{0}^{t}}\frac{1}{r}{e^{-\frac{b}{r}}}dr& \stackrel{b/r=z}{=}{\int _{b/t}^{\infty }}\frac{1}{z}{e^{-z}}\hspace{0.1667em}dz\\ {} & \le {\mathbf{1}_{\{b\ge t\}}}{\int _{b/t}^{\infty }}{e^{-z}}\hspace{0.1667em}dz+{\mathbf{1}_{\{b < t\}}}\Big({\int _{b/t}^{1}}\frac{1}{z}\hspace{0.1667em}dz+{\int _{1}^{\infty }}{e^{-z}}\hspace{0.1667em}dz\Big)\\ {} & \le {\mathbf{1}_{\{b\ge t\}}}{e^{-b/t}}+{\mathbf{1}_{\{b < t\}}}\Big(\ln \frac{t}{b}+1\Big),\end{aligned}\]We have that
To estimate $\| q(t){\| _{{B_{\alpha }^{2,2}}([j,j+1])}}$, we consider
For ${I_{1}}$, we obtain
(14)
\[ \begin{aligned}{}|{I_{1}}|\le {L_{\sigma }}{h^{\beta (\sigma )}}{\int _{0}^{t}}p(t-s,x-y)\hspace{0.1667em}ds& =C{h^{\beta (\sigma )}}{\int _{0}^{t}}\frac{{e^{-\frac{{(x-y)^{2}}}{4(t-s)}}}}{\sqrt{t-s}}\hspace{0.1667em}ds\\ {} & \stackrel{\text{(12)}}{\le }C{h^{\beta (\sigma )}}{e^{-\frac{{(x-y)^{2}}}{4T}}}.\end{aligned}\]For ${I_{2}}$, assuming $0\le h\le 1$, we get
(15)
\[ \begin{aligned}{}|{I_{2}}|& \stackrel{\text{A4}}{\le }{C_{\sigma }}{\int _{0}^{t}}|p(t-s,x-y-h)-p(t-s,x-y)|\hspace{0.1667em}ds\\ {} & ={C_{\sigma }}{\int _{0}^{t}}\Big|{\int _{x-y-h}^{x-y}}\frac{\partial p(t-s,v)}{\partial v}\hspace{0.1667em}dv\Big|\hspace{0.1667em}ds\\ {} & \stackrel{\text{(7)}}{\le }{C_{\sigma }}{C_{\lambda }}{\int _{0}^{t}}\frac{1}{t-s}{\int _{x-y-h}^{x-y}}{e^{-\frac{\lambda {v^{2}}}{t-s}}}\hspace{0.1667em}dv\hspace{0.1667em}ds\\ {} & \stackrel{\text{(11)}}{\le }C{\int _{x-y-h}^{x-y}}{e^{-\frac{\lambda {v^{2}}}{t}}}{\mathbf{1}_{\{\lambda {v^{2}}\ge t\}}}\hspace{0.1667em}dv+C{\int _{x-y-h}^{x-y}}\Big(\Big|\ln \frac{T}{\lambda {v^{2}}}\Big|+1\Big){\mathbf{1}_{\{\lambda {v^{2}} < t\}}}dv\\ {} & \le C{\int _{x-y-h}^{x-y}}{e^{-\frac{\lambda {v^{2}}}{t}}}{\mathbf{1}_{\{\lambda {v^{2}}\ge t\}}}\hspace{0.1667em}dv+C{\int _{x-y-h}^{x-y}}|\ln |v||{\mathbf{1}_{\{\lambda {v^{2}} < t\}}}dv\\ {} & \stackrel{(\ast )}{\le }C{\int _{x-y-h}^{x-y}}{e^{-\frac{\lambda {v^{2}}}{t}}}{\mathbf{1}_{\{\lambda {v^{2}}\ge t\}}}\hspace{0.1667em}dv+C{\mathbf{1}_{\{|x-y| < \sqrt{t/\lambda }+1\}}}{\int _{0}^{h/2}}|\ln z|\hspace{0.1667em}dz\\ {} & \stackrel{(\ast \ast )}{\le }Ch{e^{-\frac{\tilde{\lambda }{(x-y)^{2}}}{T}}}+C{\mathbf{1}_{\{|x-y| < \sqrt{t/\lambda }+1\}}}(z-z\ln z){\Big|_{0}^{h/2}}\\ {} & \le Ch{e^{-\frac{\tilde{\lambda }{(x-y)^{2}}}{T}}}+Ch|\ln h|{\mathbf{1}_{\{|x-y| < \sqrt{t/\lambda }+1\}}}\\ {} & \le Ch{e^{-\frac{\tilde{\lambda }{(x-y)^{2}}}{T}}}+C{h^{\beta (\sigma )}}{\mathbf{1}_{\{|x-y| < \sqrt{T/\lambda }+1\}}}\le C{h^{\beta (\sigma )}}{e^{-\frac{\tilde{\lambda }{(x-y)^{2}}}{T}}}.\end{aligned}\]In (*) we have used that the maximal value of ${\textstyle\int _{{x_{1}}}^{{x_{2}}}}|\ln |v||{\mathbf{1}_{\{|v| < C\}}}dv$ for fixed $|{x_{1}}-{x_{2}}|$ is achieved when ${x_{2}}=-{x_{1}}$.
In (**) we applied
which holds for $0<\tilde{\lambda }<\lambda $, $x-y-h\le v\le x-y$.
Further, we note that for $y\in [j,j+1]$
\[ {e^{-\frac{{(x-y)^{2}}\tilde{\lambda }}{T}}}\le {e^{\frac{(1-{(|x-j|-1)^{2}})\tilde{\lambda }}{T}}},\]
and (14) and (15) imply that
From (10), (13), and (17) we obtain (9), where
(18)
\[ \begin{aligned}{}{g^{2}}(x)& =\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}\Big({e^{\frac{2(1-{(|x-j|-1)^{2}})\tilde{\lambda }}{T}}}{\mu ^{2}}((j,j+1])\\ {} & \hspace{1em}+{e^{\frac{2(1-{(|x-j|-1)^{2}})\tilde{\lambda }}{T}}}\Big\{\sum \limits_{n\ge 1}{2^{-n(2\alpha -1)}}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}\Big).\end{aligned}\]We will check that $g\in {\mathsf{L}^{2}}(\mathbb{R})$, and get
(19)
\[ \begin{aligned}{}{\int _{\mathbb{R}}}{g^{2}}(x)\hspace{0.1667em}dx& \le C\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}{\mu ^{2}}((j,j+1])\\ {} & \hspace{1em}+C\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}\Big\{\sum \limits_{n\ge 1}{2^{-n(2\alpha -1)}}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}.\end{aligned}\]Here the sums with μ have a form ${\textstyle\sum _{l=1}^{\infty }}{\Big({\textstyle\int _{\mathbb{R}}}{f_{l}}\hspace{0.1667em}d\mu \Big)^{2}}$, where
we have $\mathsf{P}({\Omega _{\theta ,\alpha }})\hspace{-0.1667em}=\hspace{-0.1667em}1$. For each $\omega \hspace{-0.1667em}\in \hspace{-0.1667em}{\Omega _{\theta ,\alpha }}$, we have $g(\cdot ,\omega )\hspace{-0.1667em}\in \hspace{-0.1667em}{\mathsf{L}^{2}}(\mathbb{R})$, and $\widetilde{\vartheta }(\cdot ,\omega )\hspace{-0.1667em}\in \hspace{-0.1667em}\mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$. □
\[\begin{aligned}{}\{{f_{l}}(y),\hspace{2.5pt}l\ge 1\}& =\{{(|j|+1)^{\theta /2}}\hspace{0.1667em}{\mathbf{1}_{(j,j+1]}}(y),\hspace{2.5pt}j\in \mathbb{Z}\},\\ {} \{{f_{l}}(y),\hspace{2.5pt}l\ge 1\}& =\{{(|j|+1)^{\theta /2}}{2^{-n(\alpha -1/2)}}{\mathbf{1}_{{\Delta _{kn}^{(j)}}}}(y),\\ {} & \hspace{2em}j\in \mathbb{Z},\hspace{2.5pt}n\ge 1,\hspace{2.5pt}1\le k\le {2^{n}}\}.\end{aligned}\]
We have
By A5, ${\textstyle\sum _{l=1}^{\infty }}|{f_{l}}|$ is integrable w.r.t. μ on $\mathbb{R}$ in both cases. From Lemma 1 it follows that for
(20)
\[ {\Omega _{\theta ,\alpha }}=\Big\{\omega \in \Omega :\hspace{2.5pt}{\sum \limits_{l=1}^{\infty }}{\Big({\int _{\mathsf{X}}}{f_{l}}\hspace{0.1667em}d\mu \Big)^{2}}<+\infty \Big\}\]In particular, Lemma 3 implies that for each $\omega \in {\Omega _{\theta ,\alpha }}$ it holds
We will need one more boundness result about $\widetilde{\vartheta }$.
Proof.
In the proof of Lemma 3, for function g defined in (18), we obtained that estimate (9) holds. From (18) it follows that
\[\begin{aligned}{}{g^{2}}(x)& \le C\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}{\mu ^{2}}((j,j+1])\\ {} & \hspace{1em}+C\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{\theta }}\Big\{\sum \limits_{n\ge 1}{2^{-n(2\alpha -1)}}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}.\end{aligned}\]
Further, we repeat the proof of Lemma 3 after (19), and for each $\omega \in {\Omega _{\theta ,\alpha }}$ obtain that ${\sup _{x\in \mathbb{R}}}|g(x,\omega )|<\infty $. □5 Solution to the auxiliary equation
Consider the operator ${\pi _{N}}:{\mathsf{L}^{2}}(\mathbb{R})\to {\mathsf{L}^{2}}(\mathbb{R})$ such that
\[ {\pi _{N}}(v)=\left\{\begin{array}{l@{\hskip10.0pt}l}v,\hspace{1em}& \| v{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le N,\\ {} N\frac{v}{\| v{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}},\hspace{1em}& \| v{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}>N.\end{array}\right.\]
For the Hilbert space ${\mathsf{L}^{2}}(\mathbb{R})$, it is easy to check that
In this section, we study the auxiliary equation
We consider (23) for fixed $\omega \in {\Omega _{\theta ,\alpha }}$ (see (20)), and will prove the existence and uniqueness of the solution.
(23)
\[ \begin{array}{r}\displaystyle u(t,x)={\int _{\mathbb{R}}}p(t,x-y){u_{0}}(y)\hspace{0.1667em}dy\\ {} \displaystyle +{\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)f(s,y,{\pi _{N}}(u)(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} \displaystyle -{\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial p}{\partial y}(t-s,x-y)g(s,y,{\pi _{N}}(u)(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds+\widetilde{\vartheta }(t,x).\end{array}\]Let us consider the linear operators
\[\begin{aligned}{}({J_{1}}v)(t,x)& ={\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)v(s,y)\hspace{0.1667em}dy\hspace{0.1667em}ds,\\ {} ({J_{2}}w)(t,x)& ={\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial p}{\partial y}(t-s,x-y)w(s,y)\hspace{0.1667em}dy\hspace{0.1667em}ds,\end{aligned}\]
where $t\in [0,T]$, $x\in \mathbb{R}$, $v,w\in {\mathsf{L}^{\infty }}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$.We recall some lemmas from [7].
Lemma 5 (Lemma 3.1 [7]).
The operator ${J_{1}}$ is bounded from ${\mathsf{L}^{2}}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$ to $\mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$, and the following estimate holds:
where $0\le r$, $t\le T$.
(24)
\[ \| {J_{1}}v(t)-{J_{1}}v(r){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le C|t-r{|^{1/3}}{\Big({\int _{0}^{t}}\| v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds\Big)^{1/2}},\]Lemma 6 (Lemma 3.2 [7]).
The operator ${J_{2}}$ is bounded from ${\mathsf{L}^{5}}([0,T],{\mathsf{L}^{1}}(\mathbb{R}))$ to $\mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$, and the following estimates hold:
where $0\le r$, $t\le T$.
The following statement is the analogue of Proposition 4.1 [7].
Lemma 7.
Let A2 and A3 hold, and
\[ {u_{0}}\in {\mathsf{L}^{2}}(\mathbb{R}),\hspace{1em}|\widetilde{\vartheta }(t,x)|\le C(\omega ),\hspace{1em}\| \widetilde{\vartheta }(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le C(\omega )\]
for some finite constant $C(\omega )$. Then for any fixed $N>0$ and $\omega \in {\Omega _{\theta ,\alpha }}$ equation (23) has a unique solution $u\in \mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$.
Proof.
Step 1 (existence and uniqueness of the solution).
Denote
\[\begin{aligned}{}({\mathcal{A}_{1}}u)(t,x)& ={\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)f(s,y,{\pi _{N}}(u)(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds,\\ {} ({\mathcal{A}_{2}}u)(t,x)& =-{\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial p}{\partial y}(t-s,x-y)g(s,y,{\pi _{N}}(u)(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds.\end{aligned}\]
Fix $\lambda >0$. Let $\mathcal{H}$ denote the Banach space of ${\mathsf{L}^{2}}(\mathbb{R})$-valued functions $u(t,x)$ such that $u(0,x)={u_{0}}(x)$, with the norm
We will prove that operator $\mathcal{A}$ is a contraction for large λ.
\[ \| u(t){\| _{\mathcal{H}}^{2}}={\int _{0}^{T}}{e^{-\lambda t}}\| u(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}dt.\]
Define the operator $\mathcal{A}:\mathcal{H}\to \mathcal{H}$ such that
(27)
\[ (\mathcal{A}u)(t,x)={\int _{\mathbb{R}}}p(t,x-y){u_{0}}(y)\hspace{0.1667em}dy+({\mathcal{A}_{1}}u)(t,x)+({\mathcal{A}_{2}}u)(t,x)+\widetilde{\vartheta }(t,x).\]Using the inequality
Here in (**) we used the inequality for convolution $\| v\ast w{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le \| v{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\| w{\| _{{\mathsf{L}^{1}}(\mathbb{R})}}$. Further, we have
\[ \Big\| {\int _{0}^{t}}f(s)\hspace{0.1667em}ds\Big\| \le {\int _{0}^{t}}\| f(s)\| \hspace{0.1667em}ds,\]
we have that
(28)
\[ \begin{aligned}{}& \| ({\mathcal{A}_{1}}u)(t)-({\mathcal{A}_{1}}v)(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\\ {} & \hspace{1em}\le {\int _{0}^{t}}{\Big\| {\int _{\mathbb{R}}}p(t-s,x-y)(f(s,y,{\pi _{N}}(u)(s,y))-f(s,y,{\pi _{N}}(v)(s,y)))\hspace{0.1667em}dy\Big\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\hspace{0.1667em}ds\\ {} & \hspace{1em}\stackrel{\text{A2}}{\le }{\int _{0}^{t}}\Big\| {\int _{\mathbb{R}}}p(t-s,x-y)({a_{2}}(y)+L|{\pi _{N}}(u)(s,y)|+L|{\pi _{N}}(v)(s,y)|)\\ {} & \hspace{2em}\times |{\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y)|\hspace{0.1667em}dy\Big\| {_{{\mathsf{L}^{2}}(\mathbb{R})}}\hspace{0.1667em}ds\\ {} & \hspace{1em}\stackrel{(\ast \ast )}{\le }{\int _{0}^{t}}\| p(t-s,x-y){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\| ({a_{2}}(y)+L|{\pi _{N}}(u)(s,y)|+L|{\pi _{N}}(v)(s,y)|)\\ {} & \hspace{2em}\times |{\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y)|{\| _{{\mathsf{L}^{1}}(\mathbb{R})}}\hspace{0.1667em}ds.\end{aligned}\]
\[\begin{aligned}{}& \| p(t-s,x-y){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}=C{(t-s)^{-1/2}},\\ {} & \| ({a_{2}}(y)+L|{\pi _{N}}(u)(s,y)|+L|{\pi _{N}}(v)(s,y)|)|{\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y)|{\| _{{\mathsf{L}^{1}}(\mathbb{R})}}\\ {} & \hspace{1em}\le \| ({a_{2}}(y)+L|{\pi _{N}}(u)(s,y)|+L|{\pi _{N}}(v)(s,y)|{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\\ {} & \hspace{2em}\times \| {\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\\ {} & \hspace{1em}\stackrel{\text{(22)}}{\le }(\| {a_{2}}{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}+2LN)\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}.\end{aligned}\]
Applying the Hölder inequality, we get
where ${C_{N}}$ denotes some constants that may depend on N.
(29)
\[ \begin{aligned}{}\| ({\mathcal{A}_{1}}u)(t)-({\mathcal{A}_{1}}v)(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}& \le {C_{N}}{\Big({\int _{0}^{t}}{(t-s)^{-1/4}}\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\hspace{0.1667em}ds\Big)^{2}}\\ {} & \le {C_{N}}{\int _{0}^{t}}{(t-s)^{-3/4}}\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds,\end{aligned}\]For ${\mathcal{A}_{2}}$, we have
(30)
\[ \begin{aligned}{}& \| ({\mathcal{A}_{2}}u)(t)-({\mathcal{A}_{2}}v)(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\\ {} & \hspace{1em}=\Big\| {\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial p}{\partial y}(t-s,x-y)(g(s,y,{\pi _{N}}(u)(s,y))\\ {} & \hspace{2em}-g(s,y,{\pi _{N}}(v)(s,y)))\hspace{0.1667em}dy\hspace{0.1667em}ds\Big\| {_{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\\ {} & \hspace{1em}\stackrel{\text{(25)}}{\le }\Big({\int _{0}^{t}}{(t-s)^{-3/4}}\| ({b_{3}}(y)+L|{\pi _{N}}(u)(s,y)|\\ {} & \hspace{2em}+L|{\pi _{N}}(v)(s,y)|)|{\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y)|{\| _{{\mathsf{L}^{1}}(\mathbb{R})}}\hspace{0.1667em}ds\Big){^{2}}\\ {} & \hspace{1em}\le C{\int _{0}^{t}}{(t-s)^{-3/4}}\| ({b_{3}}(y)+L|{\pi _{N}}(u)(s,y)|\\ {} & \hspace{2em}+L|{\pi _{N}}(v)(s,y)|)|{\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y)|{\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds\\ {} & \hspace{1em}\le C{\int _{0}^{t}}{(t-s)^{-3/4}}\| {b_{3}}(y)+L|{\pi _{N}}(u)(s,y)|+L|{\pi _{N}}(v)(s,y)|{\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\\ {} & \hspace{2em}\times \| {\pi _{N}}(u)(s,y)-{\pi _{N}}(v)(s,y){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds\\ {} & \hspace{1em}\le C{\int _{0}^{t}}{(t-s)^{-3/4}}{(\| {b_{3}}{\| _{{\mathsf{L}^{2}}(\mathbb{R})}}+2LN)^{2}}\| u(s,y)-v(s,y){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds\\ {} & \hspace{1em}={C_{N}}{\int _{0}^{t}}{(t-s)^{-3/4}}\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds.\end{aligned}\]Therefore,
\[\begin{aligned}{}\| \mathcal{A}u-\mathcal{A}v{\| _{\mathcal{H}}^{2}}& ={\int _{0}^{T}}{e^{-\lambda t}}\| (\mathcal{A}u)(t)-(\mathcal{A}v)(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}dt\\ {} & \stackrel{\text{(29)},\text{(30)}}{\le }{C_{N}}{\int _{0}^{T}}{e^{-\lambda t}}{\int _{0}^{t}}{(t-s)^{-3/4}}\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds\hspace{0.1667em}dt\\ {} & ={C_{N}}{\int _{0}^{T}}{e^{-\lambda s}}\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}{\int _{s}^{T}}{(t-s)^{-3/4}}{e^{-\lambda (t-s)}}\hspace{0.1667em}dt\hspace{0.1667em}ds\\ {} & \le {C_{N}}{\int _{0}^{T}}{e^{-\lambda s}}\| u(s)-v(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}{\int _{0}^{\infty }}{r^{-3/4}}{e^{-\lambda r}}\hspace{0.1667em}dr\hspace{0.1667em}ds\\ {} & ={C_{N}}\| u-v{\| _{\mathcal{H}}^{2}}{\int _{0}^{\infty }}{r^{-3/4}}{e^{-\lambda r}}\hspace{0.1667em}dr,\end{aligned}\]
and for large λ we can get
Then the operator $\mathcal{A}$ on $\mathcal{H}$ is a contraction and has a unique fixed point which is the solution of (23).
Step 2 (continuity of the solution in ${\mathsf{L}^{2}}(\mathbb{R})$).
We will demonstrate that $u\in \mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$ provided that $\mathcal{A}u=u$. We consider each term in (27), and obtain
\[\begin{aligned}{}& \| ({\mathcal{A}_{1}}u)(t)-({\mathcal{A}_{1}}u)(r){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\\ {} & \hspace{1em}\stackrel{\text{(24)}}{\le }{(t-r)^{2/3}}{\int _{0}^{t}}\| f(s,y,{\pi _{N}}(v)(s,y)){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds\\ {} & \hspace{1em}\stackrel{\text{A2}}{\le }{(t-r)^{2/3}}2T(\| {a_{1}}{\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}+{K^{2}}{N^{2}}),\\ {} & \| ({\mathcal{A}_{2}}u)(t)-({\mathcal{A}_{2}}u)(r){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{5}}\stackrel{\text{(26)}}{\le }{(t-r)^{5/21}}{\int _{0}^{t}}\| g(s,y,{\pi _{N}}(v)(s,y)){\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{5}}\hspace{0.1667em}ds\\ {} & \hspace{1em}\stackrel{\text{A3}}{\le }{(t-r)^{5/21}}\cdot CT(\| {b_{1}}{\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{5}}+\| {b_{2}}{\pi _{N}}(v){\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{5}}+{K^{5}}\| {\pi _{N}^{2}}(v){\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{5}})\\ {} & \hspace{1em}\le C{(t-r)^{5/21}}(\| {b_{1}}{\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{5}}+\| {b_{2}}{\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{5}}\| {\pi _{N}}(v){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{5}}+{K^{5}}\| {\pi _{N}}(v){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{5}})\\ {} & \hspace{1em}\le C{(t-r)^{5/21}}(\| {b_{1}}{\| _{{\mathsf{L}^{1}}(\mathbb{R})}^{5}}+\| {b_{2}}{\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{5}}{N^{5}}+{K^{5}}{N^{5}}),\end{aligned}\]
therefore $({\mathcal{A}_{1}}u)(t)$ and $({\mathcal{A}_{2}}u)(t)$ are continuous. For $\tilde{\vartheta }$, we refer to Lemma 3, convolution $p\ast {u_{0}}$ is continuous by standard properties of the heat semigroup. □6 Solution to the main equation
We will use one more statement from [7].
Lemma 8 (Lemma 4.2 [7]).
Let $\zeta =\{\zeta (t,x),t\in [0,T],x\in \mathbb{R}\}$ be a continuous and bounded function belonging to $\mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$. Let $v\in \mathbb{C}([0,T],{\mathsf{L}^{2}}(\mathbb{R}))$ be a solution of the integral equation
where ${u_{0}}(y)\in {\mathsf{L}^{2}}(\mathbb{R})$, and f and g satisfy the assumptions A2 and A3.
(31)
\[ \begin{aligned}{}v(t,x)& ={\int _{\mathbb{R}}}p(t,x-y){u_{0}}(y)\hspace{0.1667em}dy\\ {} & \hspace{1em}+{\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)f(s,y,v(s,y)+\zeta (s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}-{\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial p}{\partial y}(t-s,x-y)g(s,y,v(s,y)+\zeta (s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds,\end{aligned}\]
Then we have
where the constants ${C_{1}}$ and ${C_{2}}$ depend only on T and on the functions ${a_{i}}$, ${b_{i}}$ and the constants K and L appearing in the hypothesis A2 and A3, and
(32)
\[ \| v(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\le (\| {u_{0}}{\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}+{C_{1}}(1+{R_{1}}(\zeta )))\exp \{{C_{2}}(1+{R_{2}}(\zeta ))\},\]
\[\begin{aligned}{}{R_{1}}(\zeta )& =\underset{s\in [0,T]}{\sup }(\| \zeta (s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}+\| \zeta (s){\| _{{\mathsf{L}^{4}}(\mathbb{R})}^{4}}+\| \zeta (s){\| _{{\mathsf{L}^{\infty }}(\mathbb{R})}^{2}}),\\ {} {R_{2}}(\zeta )& =\underset{s\in [0,T]}{\sup }\| \zeta (s){\| _{{\mathsf{L}^{\infty }}(\mathbb{R})}^{2}}.\end{aligned}\]
The main result of the paper is the following.
Proof.
We consider equation (5) for each fixed $\omega \in {\Omega _{\theta ,\alpha }}$, take the version $\widetilde{\vartheta }$ that satisfies (6).
We apply Lemma 8 to
\[ \zeta (t,x)={\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y)=\widetilde{\vartheta }(t,x),\]
assumptions on ζ are fulfilled by Lemmas 3 and 4.For given ζ, set
where ${N_{1}^{2}}$ is equal to the right-hand side of (32). For given N, take u which is a solution of (23).
By Lemma 8, $\| v(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le {N_{1}}$, and
\[ \| u(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le \| v(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}+\| \zeta (t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\le N-1,\hspace{1em}t\le {t_{N}}.\]
Therefore, ${t_{N}}=T$, and we never get $\| u(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}>N$. The solution of (23) will satisfy (5).7 Averaging principle
In this section, we consider equation (1), where functions f and g do not depend on the time variable t, and study the averaging of the stochastic term.
For $\varepsilon >0$, consider equation
(34)
\[ \begin{array}{r}\displaystyle \frac{\partial {u_{\varepsilon }}}{\partial t}=\frac{{\partial ^{2}}{u_{\varepsilon }}}{\partial {x^{2}}}+f(x,{u_{\varepsilon }}(t,x))+\frac{\partial g}{\partial x}(x,{u_{\varepsilon }}(t,x))+\sigma (t/\varepsilon ,x)\frac{\partial \mu }{\partial x},\\ {} \displaystyle {u_{\varepsilon }}(0,x)={u_{0}}(x).\end{array}\]Assume that for each $y\in \mathbb{R}$ there exists the limit
It is easy to see that $\bar{\sigma }(y)$ satisfies Assumption A4 with the same constants ${C_{\sigma }}$, ${L_{\sigma }}$.
(35)
\[ \bar{\sigma }(y)=\underset{t\to \infty }{\lim }\frac{1}{t}{\int _{0}^{t}}\sigma (s,y)\hspace{0.1667em}ds.\]We will study convergence ${u_{\varepsilon }}(t,x)\to \bar{u}(t,x)$, $\hspace{2.5pt}\varepsilon \to 0$, were $\bar{u}$ is the solution of the averaged equation
The mild forms of (34) and (36) are, respectively,
\[\begin{aligned}{}{u_{\varepsilon }}(t,x)& ={\int _{0}^{t}}p(t,x-y){u_{0}}(y)\hspace{0.1667em}dy\\ {} & \hspace{1em}+{\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)f(y,{u_{\varepsilon }}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}-{\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial }{\partial y}p(t-s,x-y)g(y,{u_{\varepsilon }}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}+{\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s/\varepsilon ,y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y).\end{aligned}\]
and
\[\begin{aligned}{}\bar{u}(t,x)& ={\int _{\mathbb{R}}}p(t,x-y){u_{0}}(y)\hspace{0.1667em}dy\\ {} & \hspace{1em}+{\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y)f(y,\bar{u}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}-{\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial }{\partial y}p(t-s,x-y)g(y,\bar{u}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\\ {} & \hspace{1em}+{\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\bar{\sigma }(y)\hspace{0.1667em}ds\hspace{0.1667em}d\mu (y)\hspace{0.1667em}.\end{aligned}\]
We also impose the following additional condition that is standard in the averaging principle.
Assumption A6.
The function ${G_{\sigma }}(r,y)={\textstyle\int _{0}^{r}}(\sigma (s,y)-\bar{\sigma }(y))\hspace{0.1667em}ds$, $r\in {\mathbb{R}_{+}}$, $y\in \mathbb{R}$ is bounded.
This holds, for example, if $\sigma (s,y)$ is bounded and periodic in s for each fixed y, and the set of values of minimal periods is bounded. Obviously, A6 implies (35).
Proof.
In Step 1 of the proof of Theorem 1 [19] (or Theorem 7.1 [20]), for the stochastic integrals in
\[\begin{aligned}{}{\xi _{\varepsilon }}(t,x)& ={\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\sigma (s/\varepsilon ,y)ds\hspace{0.1667em}d\mu (y)\\ {} & \hspace{1em}-{\int _{\mathbb{R}}}{\int _{0}^{t}}p(t-s,x-y)\bar{\sigma }(y)ds\hspace{0.1667em}d\mu (y)\end{aligned}\]
and some ${\gamma _{1}}>0$ it was proved that there exists a version such that
for all $\omega \in {\Omega _{1}}$, $\mathsf{P}({\Omega _{1}})=1$ (here Assumption A6 was used).As in the proof of Lemma 3, for function g defined in (18), we obtain that
where $g\in {\mathsf{L}^{2}}(\mathbb{R})$ for each $\omega \in {\Omega _{\theta ,\alpha }}$, $\mathsf{P}({\Omega _{\theta ,\alpha }})=1$, and g is independent of ε.
By (38), $|{\xi _{{\varepsilon _{n}}}}(t,x)|\to 0$, ${\varepsilon _{n}}\to 0$. The dominated convergence theorem imply that for each $t\in [0,T]$ and each $\omega \in {\Omega _{\theta ,\alpha }}\cap {\Omega _{1}}$ it holds
therefore
We can see that also
for all $\varepsilon >0$ for the same N. To explain this, note that in (33) ${N_{1}}$ depends only on T, on the functions ${a_{i}}$, ${b_{i}}$ and the constants K and L appearing in the assumptions A2 and A3. For $\zeta (t,s)=\vartheta (t,s)$, in the proof of Lemma 3 we obtained that
\[ \underset{t\in [0,T]}{\sup }\| \vartheta (t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\le C{\int _{\mathbb{R}}}{g^{2}}(x)\hspace{0.1667em}dx,\]
where the right-hand side may be estimated by (19). The constants in (19) may depend on ${C_{\sigma }}$ and ${L_{\sigma }}$ from assumption A4, but are independent of ε.Further, we obtain
\[\begin{aligned}{}& \| {u_{\varepsilon }}(t)-\bar{u}(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\\ {} & \hspace{1em}\le \Big\| {\int _{0}^{t}}{\int _{\mathbb{R}}}p(t-s,x-y){(f(y,{u_{\varepsilon }}(s,y))-f(y,\bar{u}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\Big\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\\ {} & \hspace{2em}+\Big\| {\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{\partial }{\partial y}p(t-s,x-y){(g(y,{u_{\varepsilon }}(s,y))-g(y,\bar{u}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}ds\Big\| _{{\mathsf{L}^{2}}(\mathbb{R})}}\\ {} & \hspace{2em}+\| {\xi _{\varepsilon }}(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}:={J_{1}}+{J_{2}}+\| {\xi _{\varepsilon }}(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}}.\end{aligned}\]
For ${J_{1}}$, as in (28)–(29), taking into account (40) and (41), we get
For ${J_{2}}$, as in (30), we obtain
Therefore,
We use the generalized Gronwall’s inequality (see, for example, Corollary 1 in [25]) and get
\[\begin{aligned}{}\| {u_{\varepsilon }}(t)-\bar{u}(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}& \le C(\omega )\| {\xi _{\varepsilon }}(t){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\\ {} & \hspace{1em}+C(\omega ){\int _{0}^{t}}{\sum \limits_{n=1}^{\infty }}\frac{{\Gamma ^{n}}(1/4)}{\Gamma (n/4)}{(t-s)^{n/4-1}}\| {\xi _{\varepsilon }}(s){\| _{{\mathsf{L}^{2}}(\mathbb{R})}^{2}}\hspace{0.1667em}ds.\end{aligned}\]
