1 Introduction
In this paper we establish the averaging principle for the stochastic parabolic equation
where ε is a small positive parameter, $(t,x)\in [0,T]\times \mathbb{R}$, μ is a general stochastic measure on Borel σ-algebra on $\mathbb{R}$ (see Section 2), f, σ are measurable functions, $\mathcal{L}$ is the operator of the form
Here $a,b,c$ are defined on $[0,T]$. We consider convergence ${u_{\varepsilon }}(t,x)\to \bar{u}(t,x),\varepsilon \to {0^{+}}$, where $\bar{u}$ is the solution of the averaged equation
Functions $\bar{f}$, $\bar{\sigma }$ are defined below. Note that we consider solutions to the formal equations (1) and (3) in the mild form.
(1)
\[ \left\{\begin{array}{l}\mathcal{L}{u_{\varepsilon }}(t,x)dt+f(t/\varepsilon ,x,{u_{\varepsilon }}(t,x))\hspace{0.1667em}dt+\sigma (t/\varepsilon ,x)\hspace{0.1667em}d\mu (x)=0\hspace{0.1667em},\\ {} {u_{\varepsilon }}(0,x)={u_{0}}(x)\hspace{0.1667em},\end{array}\right.\](2)
\[ \mathcal{L}u(t,x)=a(t)\frac{{\partial ^{2}}u(t,x)}{{\partial ^{2}}x}+b(t)\frac{\partial u(t,x)}{\partial x}+c(t)u(t,x)-\frac{\partial u(t,x)}{\partial t}.\](3)
\[ \left\{\begin{array}{l}\mathcal{L}\bar{u}(t,x)dt+\bar{f}(x,\bar{u}(t,x))\hspace{0.1667em}dt+\bar{\sigma }(x)\hspace{0.1667em}d\mu (x)=0\hspace{0.1667em},\\ {} \bar{u}(0,x)={u_{0}}(x)\hspace{0.1667em}.\end{array}\right.\]Averaging is widely used to describe the asymptotic behaviour of both stochastic and deterministic systems. Stochastic parabolic equation with the random noise represented by a general stochastic measure was introduced in [1]. The averaging principle for two-time-scales system driven by two independent Wiener processes was studied, for example, in [7]. Some other equations driven by Wiener process are considered in [4, 6] and [20]. Different types of equations with general stochastic measures are investigated in [2, 3, 14, 19] and [16].
The rest of the paper is organized as follows. Section 2 contains the basic facts concerning stochastic measures and integrals with respect to them. Section 3 contains precise formulation of the problem, assumptions, auxiliary statements while the main result is proved in Section 4. Some examples are given in Section 5.
2 Preliminaries
Let $(\Omega ,\mathcal{F},\mathsf{P})$ be a complete probability space and $\mathcal{B}$ be a Borel σ-algebra on $\mathbb{R}$. Denote the set of all real-valued random variables defined on $(\Omega ,\mathcal{F},\mathsf{P})$ as ${\mathsf{L}_{0}}={\mathsf{L}_{0}}(\Omega ,\mathcal{F},\mathsf{P})$, convergence in ${\mathsf{L}_{0}}$ means the convergence in probability.
Definition 1.
A σ-additive mapping $\mu :\mathcal{B}\to {\mathsf{L}_{0}}$ is called stochastic measure (SM).
In other words, μ is a vector measure with values in ${\mathsf{L}_{0}}$. We do not assume any martingale properties or moment existence for SM.
Consider some examples of SMs. If ${M_{t}}$ is a square integrable martingale then $\mu (A)={\textstyle\int _{0}^{T}}{\mathbf{1}_{A}}(t)\hspace{0.1667em}d{M_{t}}$ is an SM. α-stable random measure on $\mathcal{B}$ for $\alpha \in (0,1)\cup (1,2]$, as it is defined in [18, Sections 3.2–3.3], is an SM by Definition 1. For a fractional Brownian motion ${W_{t}^{H}}$ with Hurst index $H>1/2$ and a bounded measurable function $f:[0,T]\to \mathbb{R}$ we can define an SM $\mu (A)={\textstyle\int _{0}^{T}}f(t){\mathbf{1}_{A}}(t)\hspace{0.1667em}d{W_{t}^{H}}$, see [11, Theorem 1.1]. Some other examples can be found in [14].
In [10, Chapter 7] the definition of the integral ${\textstyle\int _{A}}g\hspace{0.1667em}d\mu $, where $g:\mathbb{R}\to \mathbb{R}$ is a deterministic measurable function, $A\in \mathcal{B}$ and μ is an SM, is given and its properties are studied. In particular, every bounded measurable g is integrable with respect to (w.r.t.) any μ. This integral was constructed and studied in [10] for μ defined on an arbitrary σ-algebra, but in our paper, we consider SM on Borel subsets of $\mathbb{R}$.
In the sequel, μ denotes a SM, C and $C(\omega )$ denote positive constant and positive random constant, respectively, whose exact values are not important ($C<\infty $, $C(\omega )<\infty $ a.s.).
We will use the following statement.
Lemma 1 (Lemma 3.1 in [12]).
Let ${\phi _{l}}:\hspace{2.5pt}\mathbb{R}\to \mathbb{R},\hspace{2.5pt}l\ge 1$, be measurable functions such that $\tilde{\phi }(x)={\textstyle\sum _{l=1}^{\infty }}|{\phi _{l}}(x)|$ is integrable w.r.t. μ on $\mathbb{R}$. Then
We consider the Besov spaces ${B_{22}^{\alpha }}([c,d])$, $0<\alpha <1$, with a standard norm
where
For any $j\in \mathbb{Z}$ and all $n\ge 0$, put
The following lemma is a key tool for estimates of the stochastic integral.
Lemma 2 (Lemma 3 in [13]).
Let Z be an arbitrary set, and the function $q(z,s):Z\times [j,j+1]\to \mathbb{R}$ is such that all paths $q(z,\cdot )$ are continuous on $[j,j+1]$. Denote
such that for all $\beta >0$, $\omega \in \Omega $, $z\in Z$
\[ {q_{n}}(z,s)=\sum \limits_{1\le k\le {2^{n}}}q\big(z,{d_{(k-1)n}^{(j)}}\big){\mathbf{1}_{{\Delta _{kn}^{(j)}}}}(s).\]
Then the random function
\[ \eta (z)={\int _{(j,j+1]}}\hspace{0.1667em}q(z,s)\hspace{0.1667em}d\mu (s),\hspace{2.5pt}z\in Z,\]
has a version
(4)
\[\begin{aligned}{}\widetilde{\eta }(z)& ={\int _{(j,j+1]}}\hspace{0.1667em}{q_{0}}(z,s)\hspace{0.1667em}d\mu (s)\\ {} & +\sum \limits_{n\ge 1}\Big({\int _{(j,j+1]}}\hspace{0.1667em}{q_{n}}(z,s)\hspace{0.1667em}d\mu (s)-{\int _{(j,j+1]}}\hspace{0.1667em}{q_{n-1}}(z,s)\hspace{0.1667em}d\mu (s)\Big)\end{aligned}\](5)
\[\begin{aligned}{}|\widetilde{\eta }(z)|\le |q(z,j)\mu ((j,j+1])|& +{\Big\{\sum \limits_{n\ge 1}{2^{n\beta }}\sum \limits_{1\le k\le {2^{n}}}|q(z,{d_{kn}^{(j)}})-q(z,{d_{(k-1)n}^{(j)}}){|^{2}}\Big\}^{1/2}}\\ {} & \times {\Big\{\sum \limits_{n\ge 1}{2^{-n\beta }}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}^{1/2}}.\end{aligned}\]Theorem 1.1 [9] implies that for $\alpha =(\beta +1)/2$,
3 Formulation of the problem and auxiliary lemmas
We consider the mild solutions to (1), i.e. the measurable random functions ${u_{\varepsilon }}(t,x)={u_{\varepsilon }}(t,x,\omega ):[0,T]\times \mathbb{R}\times \Omega \to \mathbb{R}$ such that the equations
hold a.s. for each $(t,x)\in [0,T]\times \mathbb{R}$. Here p is the fundamental solution of operator $\mathcal{L}$ from (2).
(7)
\[\begin{array}{l}\displaystyle {u_{\varepsilon }}(t,x)\hspace{0.1667em}=\hspace{0.1667em}{\int _{\mathbb{R}}}p(t,x-y;0){u_{0}}(y)\hspace{0.1667em}dy\hspace{0.1667em}+\hspace{0.1667em}{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x-y;s)f(s/\varepsilon ,y,{u_{\varepsilon }}(s,y))\hspace{0.1667em}dy\\ {} \displaystyle +{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\sigma (s/\varepsilon ,y)\hspace{0.1667em}ds\end{array}\]We will refer to the following assumptions on f, σ, ${u_{0}}$.
Assumption E1.
${u_{0}}:\mathbb{R}\times \Omega \to \mathbb{R}$ is measurable and for all $y,{y_{1}},{y_{2}}\in \mathbb{R}$
\[ |{u_{0}}(y,\omega )|\le C(\omega ),\hspace{1em}|{u_{0}}({y_{1}},\omega )-{u_{0}}({y_{2}},\omega )|\le {L_{{u_{0}}}}(\omega )|{y_{1}}-{y_{2}}{|^{\beta ({u_{0}})}},\]
where $C(\omega )$, ${L_{{u_{0}}}}(\omega )$ are random constants, $\beta ({u_{0}})\ge 1/2$.Assumption E2.
$f:{\mathbb{R}_{+}}\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ is measurable, bounded, and
\[ |f(s,{y_{1}},{v_{1}})-f(s,{y_{2}},{v_{2}})|\le {L_{f}}\big(|{y_{1}}-{y_{2}}|+|{v_{1}}-{v_{2}}|\big)\hspace{0.1667em},\]
for some constant ${L_{f}}$ and all $s\in {\mathbb{R}_{+}}$, ${y_{1}},{y_{2}},{v_{1}},{v_{2}}\in \mathbb{R}$.Assumption E3.
$\sigma :{\mathbb{R}_{+}}\times \mathbb{R}\to \mathbb{R}$ is measurable, bounded, and
for some constants ${L_{\sigma }}$, $1/2<\beta (\sigma )<1$, and all $s\in {\mathbb{R}_{+}}$, ${y_{1}},{y_{2}}\in \mathbb{R}$.
Note that if f, σ satisfy the conditions E2 and E3, respectively, $\bar{f}$ and $\bar{\sigma }$ satisfy them as well. We will show that for $\bar{f}$; the proof for $\bar{\sigma }$ is analogous. $\bar{f}$ is measurable as a limit of measurable functions, its boundedness is obvious while Lipschitz condition follows from the inequalities
\[\begin{array}{l}\displaystyle \big|\bar{f}({y_{1}},{v_{1}})-\bar{f}({y_{2}},{v_{2}})\big|=\underset{t\to \infty }{\lim }\Big|\frac{1}{t}{\int _{0}^{t}}\big(f(s,{y_{1}},{v_{1}})-f(s,{y_{2}},{v_{2}})\big)ds\Big|\\ {} \displaystyle \le \underset{t\to \infty }{\limsup }\frac{1}{t}{\int _{0}^{t}}\big|f(s,{y_{1}},{v_{1}})-f(s,{y_{2}},{v_{2}})\big|ds\le {L_{f}}\big(|{y_{1}}-{y_{2}}|+|{v_{1}}-{v_{2}}|\big).\end{array}\]
Therefore, functions
are bounded.Assertion E5 holds, for example, if $f(s,y,v)$ and $\sigma (s,y)$ are periodic in s for each y, v, and the set of values of minimal period is bounded.
[8, section 4, Theorem 1] shows that under assumption L the fundamental solution exists and
where λ and M are positive constants. Moreover, $p(t,x-y;s)$ satisfies the equation
(without loss of generality we can say that constant M in (12) is the same as in (8)–(11)). Let the stochastic measure μ satisfy the following condition.
(9)
\[ \left|\frac{\partial p(t,x-y;s)}{\partial y}\right|\le M{(t-s)^{-1}}{e^{-\frac{\lambda |x-y{|^{2}}}{t-s}}},\]
\[ a(s)\frac{{\partial ^{2}}p}{{\partial y^{2}}}-b(s)\frac{\partial p}{\partial y}+c(s)p+\frac{\partial p}{\partial s}=0\]
(see [8, (4.17)]). Using boundedness of a, b, c and estimates (8)–(10), we obtain that
(12)
\[\begin{array}{l}\displaystyle \left|\frac{\partial p(t,x-y;s)}{\partial s}\right|\le |c(s)||p(t,x-y;s)|+|b(s)|\Big|\frac{\partial p(t,x-y;s)}{\partial y}\Big|\\ {} \displaystyle +|a(s)|\Big|\frac{{\partial ^{2}}p(t,x-y;s)}{{\partial y^{2}}}\Big|\le M{(t-s)^{-3/2}}{e^{-\frac{\lambda |x-y{|^{2}}}{t-s}}}\end{array}\]Consider the mild solution of (3):
According to [1, Theorem], fulfillment of the conditions E1–E3, L and M imply that solutions of (7), (13) exist, are unique and have Hölder continuous versions on $[\tau ,T]\times [-K,K]$ for each $\tau ,K>0$. Therefore, ${u_{\varepsilon }}$ and $\bar{u}$ have continuous versions on $(0,T]\times \mathbb{R}$.
(13)
\[\begin{aligned}{}\bar{u}(t,x)={\int _{\mathbb{R}}}p(t,x-y;0){u_{0}}(y)\hspace{0.1667em}dy& +{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x-y;s)\bar{f}(y,\bar{u}(s,y))\hspace{0.1667em}dy\\ {} & +{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\bar{\sigma }(y)\hspace{0.1667em}ds\hspace{0.1667em}.\end{aligned}\]Following auxiliary lemmas are the analogues of [16, Lemma 4.1.–4.3.].
Lemma 3.
Let E3, L and M hold. Then for version (4) of
\[ \vartheta (x,t)={\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\sigma (s,y)\hspace{0.1667em}ds,\hspace{1em}t\in [0,T],\]
for any $\gamma <1/4$ there exists a random constant $C(\omega )<\infty $ a.s. (that depends on γ, is independent of x) such that
for all ${t_{1}},{t_{2}}\in [0,T]$, $x\in \mathbb{R}$.
Lemma 4.
Let Assumptions E1–E3, L, M hold, and u be a solution of equation
\[\begin{aligned}{}u(t,x)& ={\int _{\mathbb{R}}}p(t,x-y;0){u_{0}}(y)\hspace{0.1667em}dy\\ {} & +{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x-y;s)f(s,y,u(s,y))\hspace{0.1667em}dy\\ {} & +{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\sigma (s,y)\hspace{0.1667em}ds\hspace{0.1667em}.\end{aligned}\]
Then for the continuous version of u, each $0<\gamma <1/4$, some $C(\omega )$, and all $0<{t_{1}}<{t_{2}}\le T$, $x\in \mathbb{R}$, it holds
These lemmas are proved similarly to corresponding lemmas in [16]. We refer to (9), (11) instead of [16, (3.2)] and use (8) to prove the analogues of [16, (4.2), (4.3)].
Lemma 5.
Let $h(r,y,z):{\mathbb{R}_{+}}\times \mathbb{R}\times Z\to \mathbb{R}$, $\bar{h}(y,z):\mathbb{R}\times Z\to \mathbb{R}$ be measurable for each fixed z, the functions
be bounded on ${\mathbb{R}_{+}}\times \mathbb{R}\times Z$, the functions ${\varphi _{1}},{\varphi _{2}}$ be measurable on $[0,T]$ and the inequality $0\le {\varphi _{1}}(t)\le {\varphi _{2}}(t)\le t$ hold on $[0,T]$. Then
for all $x\in \mathbb{R}$, $t\in [0,T]$, $\varepsilon >0$, where the constant C does not depend on ${\varphi _{1}},{\varphi _{2}}$.
(15)
\[ \Big|{\int _{\mathbb{R}}}dy{\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)}}p(t,x-y;s)H(s/\varepsilon ,y,z)ds\Big|\le C\varepsilon |\ln \varepsilon |,\]Proof.
Assume that ${\varphi _{2}}(t)-{\varphi _{1}}(t)\ge \varepsilon $. In this case we can rewrite the inner integral as following: ${\textstyle\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)}}={\textstyle\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)-\varepsilon }}+{\textstyle\int _{{\varphi _{2}}(t)-\varepsilon }^{{\varphi _{2}}(t)}}$. We have that
On the other hand,
(16)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \Big|{\int _{\mathbb{R}}}dy{\int _{{\varphi _{2}}(t)-\varepsilon }^{{\varphi _{2}}(t)}}p(t,x-y;s)H(s/\varepsilon ,y,z)ds\Big|\\ {} & & \displaystyle \hspace{1em}\le C{\int _{{\varphi _{2}}(t)-\varepsilon }^{{\varphi _{2}}(t)}}ds{\int _{\mathbb{R}}}|p(t,x-y;s)|dy\le C\varepsilon .\end{array}\]
\[\begin{array}{l}\displaystyle \Big|{\int _{\mathbb{R}}}dy{\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)-\varepsilon }}p(t,x-y;s)H(s/\varepsilon ,y,z)ds\Big|\\ {} \displaystyle =C\varepsilon \bigg|{\int _{\mathbb{R}}}\Big(p(t,x-y;s)G(s/\varepsilon ,y,z){\Big|_{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)-\varepsilon }}\\ {} \displaystyle -{\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)-\varepsilon }}\frac{\partial p(t,x-y;s)}{\partial s}G(s/\varepsilon ,y,z)ds\Big)dy\bigg|\le C\varepsilon \bigg({\int _{\mathbb{R}}}\big|p(t,x-y;{\varphi _{1}}(t))\big|dy\\ {} \displaystyle +{\int _{\mathbb{R}}}\big|p(t,x-y;{\varphi _{2}}(t)-\varepsilon )\big|dy+{\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)-\varepsilon }}ds{\int _{\mathbb{R}}}\frac{dy}{{(t-s)^{3/2}}}{e^{-\frac{\lambda |x-y{|^{2}}}{t-s}}}\bigg)\\ {} \displaystyle \le C\varepsilon +C\varepsilon {\int _{{\varphi _{1}}(t)}^{{\varphi _{2}}(t)-\varepsilon }}\frac{ds}{t-s}=C\varepsilon \big(1-\ln (\varepsilon +t-{\varphi _{2}}(t))+\ln (t-{\varphi _{1}}(t))\big)\\ {} \displaystyle \le C\varepsilon \big(1+|\ln T|\vee |\ln \varepsilon |\big)\le C\varepsilon |\ln \varepsilon |,\end{array}\]
and we get (15). If ${\varphi _{2}}(t)-{\varphi _{1}}(t)<\varepsilon $, we obtain (15) similarly to (16). □The following lemma is an analogue of [15, Lemma 3].
Lemma 6.
Let $h(r,y):{\mathbb{R}_{+}}\times \mathbb{R}\to \mathbb{R}$, $\bar{h}(y):\mathbb{R}\to \mathbb{R}$ be measurable, the functions
be bounded on ${\mathbb{R}_{+}}\times \mathbb{R}$. Then
for all $x\in \mathbb{R}$, $t\in [0,T]$, $\varepsilon >0$.
Proof.
If $t\ge \varepsilon $, we can use the decomposition ${\textstyle\int _{0}^{t}}={\textstyle\int _{0}^{t-\varepsilon }}+{\textstyle\int _{t-\varepsilon }^{t}}$.
For the first summand,
For the second summand,
From (18), (19) we obtain (17). If $t<\varepsilon $, (17) is a corollary of (19). □
(18)
\[\begin{array}{l}\displaystyle \Big|{\int _{0}^{t-\varepsilon }}p(t,x-y;s)H(s/\varepsilon ,y)ds\Big|=\varepsilon \Big|p(t,x-y;s)G(s/\varepsilon ,y){\Big|_{0}^{t-\varepsilon }}\\ {} \displaystyle -{\int _{0}^{t-\varepsilon }}\frac{\partial p(t,x-y;s)}{\partial s}G(s/\varepsilon ,y)ds\Big|\stackrel{G(0,y)=0}{\le }C\varepsilon \bigg(|p(t,x-y;t-\varepsilon )|\\ {} \displaystyle +{\int _{0}^{t-\varepsilon }}\Big|\frac{\partial p(t,x-y;s)}{\partial s}\Big|ds\bigg)\stackrel{\text{(8)},\hspace{2.5pt}\text{(12)}}{\le }C\sqrt{\varepsilon }+C\varepsilon {\int _{0}^{t-\varepsilon }}ds{(t-s)^{-3/2}}\le C\sqrt{\varepsilon }.\end{array}\](19)
\[ \Big|{\int _{t-\varepsilon }^{t}}p(t,x-y;s)H(s/\varepsilon ,y)ds\Big|\le C{\int _{t-\varepsilon }^{t}}{(t-s)^{-1/2}}ds=C\sqrt{\varepsilon }.\]4 The main result
We are ready to formulate the main result of the paper.
Theorem 1.
Proof.
For each $(t,x)\in [0,T]\times \mathbb{R}$ we take versions of stochastic integrals ${\textstyle\int _{\mathbb{R}}}d\mu (y){\textstyle\int _{0}^{t}}p(t,x-y;s)\sigma (s/\varepsilon ,y)\hspace{0.1667em}ds$ and ${\textstyle\int _{\mathbb{R}}}d\mu (y){\textstyle\int _{0}^{t}}p(t,x-y;s)\bar{\sigma }(y)\hspace{0.1667em}ds$ that are defined by Lemma 2. We obtain
where
(recall that $0<\gamma <1/4$). Note that for each $k\in \{1,\dots ,n-1\}$ the function
where ${\lambda _{1}}>0$ (see (4.4) and the proof of (4.64) in [8]). Using that
On the other hand,
Denote ${\gamma _{2}}=\gamma \wedge {\lambda _{1}}$. From (23), (24), (25), (26) it follows that $|\bar{u}(t,x)-\bar{u}(0,x)|\le C{t^{{\gamma _{2}}}}$. Therefore,
(see formula 4.10 from [17]). Here we used the inequality
For ${\gamma _{1}}<\frac{1}{2}\Big(1-\frac{1}{2\beta (\sigma )}\Big)$ we can choose $\theta =2{\gamma _{1}}$, $1>\alpha >1/2$ such that (30) holds.
(21)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle |{u_{\varepsilon }}(t,x)-\bar{u}(t,x)|\\ {} & & \displaystyle \hspace{1em}\le \Big|{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x-y;s)(f(s/\varepsilon ,y,{u_{\varepsilon }}(s,y))-f(s/\varepsilon ,y,\bar{u}(s,y))\hspace{0.1667em}dy\Big|\\ {} & & \displaystyle \hspace{2em}+\Big|{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x-y;s)(f(s/\varepsilon ,y,\bar{u}(s,y))-\bar{f}(y,\bar{u}(s,y)))\hspace{0.1667em}dy\Big|+|{\xi _{\varepsilon }}|\\ {} & & \displaystyle \hspace{1em}=:{I_{1}}+{I_{2}}+|{\xi _{\varepsilon }}|,\end{array}\]
\[\begin{array}{l}\displaystyle {\xi _{\varepsilon }}={\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\sigma (s/\varepsilon ,y)ds\\ {} \displaystyle -{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\bar{\sigma }(y)ds.\end{array}\]
To estimate the second term, we divide $[0,T]$ into n segments of length $\Delta =T/n$ and rewrite ${I_{2}}$ as
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {I_{2}}=\Big|{\sum \limits_{k=0}^{n-1}}{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}ds{\int _{\mathbb{R}}}p(t,x-y;s)\\ {} & & \displaystyle \hspace{1em}\times \big(f(s/\varepsilon ,y,\bar{u}(s,y))-\bar{f}(y,\bar{u}(s,y))\big)\hspace{0.1667em}dy\Big|.\end{array}\]
Thus,
\[\begin{aligned}{}{I_{2}}& \le {\sum \limits_{k=0}^{n-1}}\bigg(\Big|{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}ds{\int _{\mathbb{R}}}p(t,x-y;s)\big(f(s/\varepsilon ,y,\bar{u}(s,y))\\ {} & -f(s/\varepsilon ,y,\bar{u}(k\Delta ,y))\big)\hspace{0.1667em}dy\Big|\\ {} & +\Big|{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}ds{\int _{\mathbb{R}}}p(t,x-y;s)\big(f(s/\varepsilon ,y,\bar{u}(k\Delta ,y))\\ {} & -\bar{f}(y,\bar{u}(k\Delta ,y))\big)\hspace{0.1667em}dy\Big|\\ {} & +\Big|{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}ds{\int _{\mathbb{R}}}p(t,x-y;s)\big(\bar{f}(y,\bar{u}(k\Delta ,y))\\ {} & -\bar{f}(y,\bar{u}(s,y))\big)\hspace{0.1667em}dy\Big|\bigg)\\ {} & =:{\sum \limits_{k=0}^{n-1}}({I_{21}^{(k)}}+{I_{22}^{(k)}}+{I_{23}^{(k)}}).\end{aligned}\]
Applying Lemma 5 for $h(r,y,z)=f(r,y,\bar{u}(z,y))$, $\bar{h}(y,z)=\bar{f}(y,\bar{u}(z,y))$, ${\varphi _{1}}(t)=k\Delta \wedge t$, ${\varphi _{2}}(t)=(k+1)\Delta \wedge t$, $k\in \{0,\dots ,n-1\}$, we obtain
We estimate ${\textstyle\sum _{k=1}^{n-1}}{I_{21}^{(k)}}$ and ${I_{21}^{(0)}}$ separately. For ${\textstyle\sum _{k=1}^{n-1}}{I_{21}^{(k)}}$ we use Lemma 4:
(22)
\[\begin{array}{l}\displaystyle {\sum \limits_{k=1}^{n-1}}{I_{21}^{(k)}}\le {L_{f}}{\sum \limits_{k=1}^{n-1}}{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}ds{\int _{\mathbb{R}}}|p(t,x-y;s)||\bar{u}(s,y)-\bar{u}(k\Delta ,y)|\hspace{0.1667em}dy\\ {} \displaystyle \le {L_{f}}C(\omega ){\sum \limits_{k=1}^{n-1}}{\int _{\mathbb{R}}}dy{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}|p(t,x-y;s)|(\ln s-\ln k\Delta \\ {} \displaystyle +s\ln s-k\Delta \ln k\Delta -(s-k\Delta )\ln (s-k\Delta )+{(s-k\Delta )^{\gamma }})ds.\end{array}\]
\[ {f_{k}}(s)=\ln s-\ln k\Delta +s\ln s-k\Delta \ln k\Delta -(s-k\Delta )\ln (s-k\Delta )+{(s-k\Delta )^{\gamma }}\]
is increasing on $[k\Delta ,(k+1)\Delta ]$. Therefore, we can estimate the sum in (22) in the following way:
\[\begin{array}{l}\displaystyle {\sum \limits_{k=1}^{n-1}}{\int _{\mathbb{R}}}dy{\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}|p(t,x-y;s)|{f_{k}}(s)\hspace{0.1667em}ds\\ {} \displaystyle \le {\sum \limits_{k=1}^{n-1}}{f_{k}}((k+1)\Delta ){\int _{(k\Delta \wedge t,(k+1)\Delta \wedge t]}}ds{\int _{\mathbb{R}}}|p(t,x-y;s)|\hspace{0.1667em}dy\\ {} \displaystyle \le C{\sum \limits_{k=1}^{n-1}}{f_{k}}((k+1)\Delta )\Delta =C\Delta {\sum \limits_{k=1}^{n-1}}\Big((\ln (k+1)\Delta -\ln k\Delta )\\ {} \displaystyle +((k+1)\Delta \ln (k+1)\Delta -k\Delta \ln k\Delta )-\Delta \ln \Delta +{\Delta ^{\gamma }}\Big)\\ {} \displaystyle =C\Delta \Big(\ln n\Delta -\ln \Delta +n\Delta \ln n\Delta -\Delta \ln \Delta -(n-1)\Delta \ln \Delta +(n-1){\Delta ^{\gamma }}\Big)\\ {} \displaystyle \stackrel{n\Delta =T}{=}CT(T+1)\frac{\ln n}{n}+\frac{n-1}{n}T{\Big(\frac{T}{n}\Big)^{\gamma }}\le C{n^{-\gamma }}.\end{array}\]
Now we need to estimate
\[\begin{aligned}{}|\bar{u}(t,x)-\bar{u}(0,x)|& =|\bar{u}(t,x)-{u_{0}}(x)|\\ {} & \le \Big|{\int _{\mathbb{R}}}p(t,x-y;0){u_{0}}(y)\hspace{0.1667em}dy-{u_{0}}(x)\Big|\\ {} & +\Big|{\int _{0}^{t}}ds{\int _{\mathbb{R}}}p(t,x-y;s)\bar{f}(y,\bar{u}(s,y))\hspace{0.1667em}dy\Big|\\ {} & +\Big|{\int _{\mathbb{R}}}d\mu (y){\int _{0}^{t}}p(t,x-y;s)\bar{\sigma }(y)\hspace{0.1667em}ds\Big|=:{I_{211}^{(0)}}+{I_{212}^{(0)}}+{I_{213}^{(0)}}.\end{aligned}\]
Note that
(23)
\[ {I_{211}^{(0)}}\le C{t^{{\lambda _{1}}}}+\Big|{\int _{\mathbb{R}}}\frac{1}{\sqrt{4\pi t\hspace{0.1667em}a(0)}}{e^{-\frac{{(x-y)^{2}}}{4ta(0)}}}{u_{0}}(y)dy-{u_{0}}(x)\Big|,\]
\[ {\int _{\mathbb{R}}}\frac{dy}{\sqrt{4\pi t\hspace{0.1667em}a(0)}}{e^{-\frac{{(x-y)^{2}}}{4ta(0)}}}=1,\]
we obtain
(24)
\[\begin{array}{l}\displaystyle \Big|{\int _{\mathbb{R}}}\frac{1}{\sqrt{4\pi t\hspace{0.1667em}a(0)}}{e^{-\frac{{(x-y)^{2}}}{4ta(0)}}}{u_{0}}(y)dy-{u_{0}}(x)\Big|\le {\int _{\mathbb{R}}}\frac{\big|{u_{0}}(y)-{u_{0}}(x)\big|}{\sqrt{4\pi t\hspace{0.1667em}a(0)}}{e^{-\frac{{(x-y)^{2}}}{4ta(0)}}}dy\\ {} \displaystyle \le C{t^{-1/2}}{\int _{\mathbb{R}}}|x-y{|^{\beta ({u_{0}})}}{e^{-\frac{{(x-y)^{2}}}{4ta(0)}}}dy\\ {} \displaystyle \stackrel{v=(x-y)/2\sqrt{ta(0)}}{=}C{t^{\beta ({u_{0}})}}{\int _{0}^{+\infty }}{e^{-{v^{2}}}}{v^{\beta ({u_{0}})}}dv=C{t^{\beta ({u_{0}})}}.\end{array}\]
\[\begin{array}{l}\displaystyle {I_{21}^{(0)}}\le {L_{f}}{\int _{(0,\Delta \wedge t]}}ds{\int _{\mathbb{R}}}|p(t,x-y;s)||\bar{u}(s,y)-\bar{u}(0,y)|\hspace{0.1667em}dy\\ {} \displaystyle \le C{\int _{(0,\Delta \wedge t]}}{s^{{\gamma _{2}}}}\hspace{0.1667em}ds\le C{n^{-1-{\gamma _{2}}}},\end{array}\]
and we can see that ${\textstyle\sum _{k=0}^{n-1}}{I_{21}^{(k)}}\le C(\omega ){n^{-\gamma }}$, where $\gamma <1/4$. Using similar arguments we can prove that ${\textstyle\sum _{k=0}^{n-1}}{I_{23}^{(k)}}\le C(\omega ){n^{-\gamma }}$. Thus we obtain
Function $g(x)=x|\varepsilon \ln \varepsilon |+{x^{-\gamma }}$, $x>0$, has the minimum value
\[ g({x_{\ast }})=|\varepsilon \ln \varepsilon {|^{\gamma /(\gamma +1)}}(\gamma +1){\gamma ^{-\gamma /(\gamma +1)}},\hspace{1em}{x_{\ast }}={(\gamma /|\varepsilon \ln \varepsilon |)^{1/(\gamma +1)}}.\]
We have $\frac{g(x+1)}{g(x)}\le C$, therefore there exists a positive integer ${n_{\ast }}=[{x_{\ast }}]+1$ such that
Recall that ${\gamma _{1}}<1/5$. Therefore, we can take $\gamma <1/4$ such that $\gamma /(\gamma +1)>{\gamma _{1}}$ and obtain
Now we estimate ${\xi _{\varepsilon }}$. Denote
We will estimate $\| q(z,\cdot ){\| _{{B_{22}^{\alpha }}((j,j+1])}}$. Consider
\[\begin{aligned}{}q& (z,y+h)-q(z,y)={J_{1}}+{J_{2}}:=\\ {} & ={\int _{0}^{t}}p(t,x-y;s)\big(\sigma (s/\varepsilon ,y+h)-\sigma (s/\varepsilon ,y)-\bar{\sigma }(y+h)+\bar{\sigma }(y)\big)\hspace{0.1667em}ds\\ {} & +{\int _{0}^{t}}\big(p(t,x-y-h;s)-p(t,x-y;s)\big)\big(\sigma (s/\varepsilon ,y+h)-\sigma (s/\varepsilon ,y)\big)\hspace{0.1667em}ds.\end{aligned}\]
Using E3, we obtain
\[ |{J_{1}}|\le 2{L_{\sigma }}{h^{\beta (\sigma )}}{\int _{0}^{t}}|p(t,x-y;s)|ds\le C{h^{\beta (\sigma )}}.\]
For ${J_{2}}$ we use boundedness of σ:
(28)
\[\begin{array}{l}\displaystyle |{J_{2}}|\le C{\int _{0}^{t}}|p(t,x-y-h;s)-p(t,x-y;s)|ds\\ {} \displaystyle =C{\int _{0}^{t}}\Big|{\int _{x-y-h}^{x-y}}\frac{\partial p(t,v;s)}{\partial v}\hspace{0.1667em}dv\Big|ds\stackrel{\text{(9)}}{\le }C{\int _{0}^{t}}\frac{ds}{t-s}{\int _{x-y-h}^{x-y}}{e^{-\frac{\lambda {v^{2}}}{t-s}}}dv\le \\ {} \displaystyle C{\int _{-h/2}^{h/2}}dr{\int _{0}^{t}}\frac{1}{t-s}{e^{-\frac{\lambda {r^{2}}}{t-s}}}ds\stackrel{z=\lambda {r^{2}}/(t-s)}{=}C{\int _{-h/2}^{h/2}}dr{\int _{\lambda {r^{2}}/t}^{\infty }}\frac{{e^{-z}}}{z}dz\\ {} \displaystyle \le C{\int _{-h/2}^{h/2}}dr\Big({\int _{\lambda {r^{2}}/t}^{1}}\frac{1}{z}dz+{\int _{1}^{\infty }}{e^{-z}}dz\Big)\le C{\int _{-h/2}^{h/2}}(1+|\ln |r||)\hspace{0.1667em}dr\\ {} \displaystyle =Ch+2C(r-r\ln r){\Big|_{0}^{h/2}}\le C{h^{\beta (\sigma )}}\end{array}\]
\[ {\int _{x-y-h}^{x-y}}{e^{-\frac{\lambda {v^{2}}}{t-s}}}dv\le {\int _{-h/2}^{h/2}}{e^{-\frac{\lambda {r^{2}}}{t-s}}}dr.\]
On the other hand, Lemma 6 implies that
From (28) and (29) it follows that for all $\theta \in [0,1]$
\[\begin{array}{l}\displaystyle |q(z,y+h)-q(z,y)|\le C{h^{\beta (\sigma )(1-\theta )}}{\varepsilon ^{\theta /2}},\\ {} \displaystyle {({\omega _{2}}(q,r))^{2}}\le C{h^{2\beta (\sigma )(1-\theta )}}{\varepsilon ^{\theta }},\\ {} \displaystyle \| q(z,\cdot ){\| _{{L_{2}}((j,j+1])}}\le C\sqrt{\varepsilon }\le C{\varepsilon ^{\theta /2}},\end{array}\]
and
if integral
is finite. That holds true if and only if
(30)
\[ \alpha <\beta (\sigma )(1-\theta )\Leftrightarrow \theta <1-\frac{\alpha }{\beta (\sigma )}.\]Using Lemma 2, we get
\[\begin{array}{l}\displaystyle |{\xi _{\varepsilon }}|=\Big|{\int _{\mathbb{R}}}q(z,y)\hspace{0.1667em}d\mu (y)\Big|\le \sum \limits_{j\in \mathbb{Z}}\Big|{\int _{(j,j+1]}}q(z,y)\hspace{0.1667em}d\mu (y)\Big|\\ {} \displaystyle \stackrel{\text{(5)},\hspace{2.5pt}\text{(6)}}{\le }\sum \limits_{j\in \mathbb{Z}}|q(z,j)\mu ((j,j+1])|\\ {} \displaystyle +C\sum \limits_{j\in \mathbb{Z}}\| q(z,\cdot ){\| _{{B_{22}^{\alpha }}([j,j+1])}}{\Big\{\sum \limits_{n\ge 1}{2^{n(1-2\alpha )}}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}^{1/2}}\\ {} \displaystyle \le C{\varepsilon ^{{\gamma _{1}}}}\Big[\sum \limits_{j\in \mathbb{Z}}|\mu ((j,j+1])|+\sum \limits_{j\in \mathbb{Z}}{\Big\{\sum \limits_{n\ge 1}{2^{n(1-2\alpha )}}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big\}^{1/2}}\Big]\\ {} \displaystyle \le C{\varepsilon ^{{\gamma _{1}}}}\Big[\Big(\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{2\rho }}{(\mu {((j,j+1]))^{2}}\Big)^{1/2}}{\Big(\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{-2\rho }}\Big)^{1/2}}\\ {} \displaystyle +{\Big(\sum \limits_{n\ge 1}{2^{n(1-2\alpha )}}\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{2\rho }}\sum \limits_{1\le k\le {2^{n}}}|\mu ({\Delta _{kn}^{(j)}}){|^{2}}\Big)^{1/2}}{\Big(\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{-2\rho }}\Big)^{1/2}}\Big],\end{array}\]
where $\rho >1/2$ is taken from Assumption M, the sums with SMs have the form ${\textstyle\sum _{l=1}^{\infty }}{\Big({\textstyle\int _{\mathbb{R}}}{\phi _{l}}\hspace{0.1667em}d\mu \Big)^{2}}$, where
\[\begin{array}{l}\displaystyle \{{\phi _{l}}(y),\hspace{2.5pt}l\ge 1\}=\{{(|j|+1)^{\rho }}\hspace{0.1667em}{\mathbf{1}_{(j,j+1]}}(y),\hspace{2.5pt}j\in \mathbb{Z}\},\\ {} \displaystyle \{{\phi _{l}}(y),\hspace{2.5pt}l\ge 1\}=\{{(|j|+1)^{\rho }}{2^{n(1-2\alpha )/2}}{\mathbf{1}_{{\Delta _{kn}^{(j)}}}}(y),\hspace{2.5pt}j\in \mathbb{Z},\hspace{2.5pt}n\ge 1,\hspace{2.5pt}1\le k\le {2^{n}}\}.\end{array}\]
From the inequalities
\[ {\sum \limits_{l=1}^{\infty }}|{\phi _{l}}(y)|\le C(1+|y{|^{\rho }}),\hspace{1em}\sum \limits_{j\in \mathbb{Z}}{(|j|+1)^{-2\rho }}<\infty ,\]
and Lemma 1 it follows that
From (21), (27) and (31) it follows that
\[\begin{aligned}{}& |{u_{\varepsilon }}(t,x)-\bar{u}(t,x)|\le C{\varepsilon ^{{\gamma _{1}}}}\\ {} & \hspace{2em}+{\int _{0}^{t}}ds{\int _{\mathbb{R}}}|p(t,x-y;s)||f(s/\varepsilon ,y,{u_{\varepsilon }}(s,y))-f(s/\varepsilon ,y,\bar{u}(s,y))\hspace{0.1667em}dy\hspace{0.1667em}|\hspace{1em}\text{a.s.}\end{aligned}\]
From boundedness of f it follows that $\underset{x\in \mathbb{R}}{\sup }|{u_{\varepsilon }}(t,x)-\bar{u}(t,x)|<\infty \hspace{2.5pt}\text{a.s.}$ On the other hand, using (8) and the Lipschitz condition on f, we get
\[ \underset{x\in \mathbb{R}}{\sup }|{u_{\varepsilon }}(t,x)-\bar{u}(t,x)|\le C{\varepsilon ^{{\gamma _{1}}}}+C{\int _{0}^{t}}\underset{x\in \mathbb{R}}{\sup }|{u_{\varepsilon }}(t,x)-\bar{u}(t,x)|\hspace{0.1667em}ds.\]
From the Gronwall inequality we obtain
\[ \underset{x\in \mathbb{R}}{\sup }|{u_{\varepsilon }}(t,x)-\bar{u}(t,x)|\le C(\omega ){\varepsilon ^{{\gamma _{1}}}},\]
where $C(\omega )$ is independent of t and ε. Thus, we obtain (20). □5 Examples
Example 1.
Let $a(t)={a^{2}}>0$, $b(t)=c(t)=0$ for each $t\in [0,T]$. Then
\[ \mathcal{L}u(t,x)={a^{2}}\frac{{\partial ^{2}}u(t,x)}{{\partial ^{2}}x}-\frac{\partial u(t,x)}{\partial t},\]
and (1), (3) are the heat equations. Note that the averaging principle for the heat equation was considered in [16], and the same order of strong convergence was obtained.Example 2.
Let $a(t)=1$, $b(t)=0$, $c(t)=-1$, $t\in [0,T]$. Then
\[ \mathcal{L}u(t,x)=\frac{{\partial ^{2}}u(t,x)}{{\partial ^{2}}x}-u(t,x)-\frac{\partial u(t,x)}{\partial t},\]
and (1), (3) are the so-called cable equations. The cable equation describes potential changes along the branch of the dendritic tree (see [5]). The averaging principle for the cable equation on $[0,L]$ with $f\equiv 0$ was established in [2] with the better order of strong convergence (${\gamma _{1}}<\frac{1}{2}(1-\frac{1}{2\beta (\sigma )})$).
