1 Introduction
Many diffusion phenomena in various fields of real life are modelled by the following type of partial differential equations (PDEs)
where $\mathcal{L}$ is the elliptic divergence form operator defined by
R and r are two measurable and bounded functions defined on $\mathbb{R}$ and satisfying
${a_{i}},{\rho _{i}}$ ($i=1,2$) are strictly positive constants. Operators of the kind (3) are the infinitesimal generators of diffusion processes that have been widely studied in literature (see [7, 12] and references therein). The discontinuity of the coefficients A and ρ reflects the heterogeneity of the media in which the modelled process under study propagates.
\[ {\mu _{1}}\le R(x)\hspace{1em}\text{and}\hspace{1em}{\mu _{2}}\le r(x)\hspace{1em}\text{for all}\hspace{2.5pt}x\in \mathbb{R}\]
where ${\mu _{1}}$ and ${\mu _{2}}$ are two strictly positive real constants, and $\frac{d}{dx}$ denotes the derivative in the distributional sense. More information on PDEs of the type (1) and their applications can be found, e.g., in [5, 15, 19] and references therein. One interesting example of such PDEs is the one defined by
(4)
\[ A(x)={a_{1}}{\mathbf{1}_{\{x\le 0\}}}+{a_{2}}{\mathbf{1}_{\{0<x\}}}\hspace{1em}\text{and}\hspace{1em}\rho (x)={\rho _{1}}{\mathbf{1}_{\{x\le 0\}}}+{\rho _{2}}{\mathbf{1}_{\{0<x\}}},\]In the present paper, we introduce a stochastic partial differential equation (SPDE), that can be considered as a stochastic counterpart of PDE (1). More specifically, we consider
where $\mathcal{L}$ is defined by (2) and $\dot{W}$ denotes the formal derivative of a space-time white noise. That is, W is a centered Gaussian field $W=\{W(t,C);t\in [0,T],C\in {B_{b}}(\mathbb{R})\}$ with covariance
where λ denotes the Lebesgue measure and ${B_{b}}(\mathbb{R})$ is the set of λ-bounded Borel sub-sets of $\mathbb{R}$. So W behaves as a Wiener process both in time and in space. The solution to Equation (5) is a random field $\{u(t,x),t\ge 0,x\in \mathbb{R}\}$, where t represents the time variable and x is the space variable. In the particular case where the functions r and R are constants $r:=2$ and $R:=1$, the operator $\mathcal{L}$ is reduced to $\frac{1}{2}\frac{{\partial ^{2}}}{\partial {x^{2}}}$. So Equation (5) also represents a natural extension of the stochastic heat equation driven by the space-time white noise, which has been widely studied in the literature (see [22] and the references therein). This can be considered as an important motivation for the investigation of such equation’s solution.
(5)
\[ \left\{\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \frac{\partial u(t,x)}{\partial t}& =& \mathcal{L}u(t,x)+\dot{W}(t,x);\hspace{1em}t>0,x\in \mathbb{R},\\ {} u(0,.)& :=& 0,\hspace{0.2778em}\end{array}\right.\]This paper has a twofold objective: the first is to lay the first milestone towards the investigation of the stochastic process solution to (5). We prove its existence, and we investigate its spatial quadratic variation. In fact, the study of quadratic variation is motivated by its numerous applications in many fields. For example, in the estimation theory, the analysis of the asymptotic behaviour of the quadratic variations of self-similar processes play an important role in the construction of consistent estimators for the self-similarity parameter (see, e.g., [24] and references therein). In stochastic analysis, quadratic variations are as well one of the main tools used to characterize the semi-martingale property for some mixed Gaussian processes (see, e.g., [10, 14, 29]). Examples of applications of quadratic variation investigation include also the theory of the Itô stochastic calculus with respect to martingales [21] and mathematical finance [3]. We refer to the monograph [22] for a more complete exposition on variations of stochastic processes in general, and of solutions to certain SPDEs in particular. In this paper, under some conditions on the fundamental solution to PDE (1), we fix t and study the limit behaviour in distribution of the sequence ${({\textstyle\sum _{j=0}^{N-1}}{(u(t,\frac{j+1}{N})-u(t,\frac{j}{N}))^{2}})_{N\ge 1}}$. More precisely, using some elements of the Stein–Malliavin calculus, we show that, after recentralization and renormalization, the above sequence satisfies an Almost Sure Central Limit Theorem (in short: ASCLT). Similar study has been done in the case of stochastic heat equation (see [20, 23]) and also in the case of stochastic wave equation (see [11]). But no similar study has been carried out on SPDEs (5). For more information on ASCLT, see [2] and references therein. The second objective of this paper is to make a further study of the SPDE defined by
where ${\mathcal{L}_{p}}$ is the operator defined by (3). We note that Equation (7) is a particular case of (5), and it could be a good model for diffusion phenomena in a medium consisting of two kind of materials, undergoing stochastic perturbations. Equation (7) has been introduced in [30] where the authors proved the existence of the solution and they presented explicit expressions of its covariance and variance functions. Some regularity properties of the solution sample paths have also been analyzed. In [31], Zili and Zougar expanded the quartic variations in time and the quadratic variations in space of the solution to Equation (7). Both expansions allowed them to deduce an estimation method of the parameters ${a_{1}}$ and ${a_{2}}$ appearing in (4). We make here another step in the study of SPDE (7) by showing that its solution satisfies all conditions under which we can use the ASCLT. In addition to the Stein–Malliavin calculus, our proofs require many integration techniques, calculation, and analysis tools.
(7)
\[ \left\{\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \frac{\partial u(t,x)}{\partial t}& =& {\mathcal{L}_{p}}u(t,x)+\dot{W}(t,x);\hspace{1em}t>0,x\in \mathbb{R},\\ {} u(0,.)& :=& 0,\hspace{0.2778em}\end{array}\right.\]The paper is organized as follows. In the next section, we prove the existence of the mild solution to Equation (5) and we give some characterizations of its spatial increments. In Section 3, using some elements of the Stein–Malliavin theory, we establish an almost sure central limit theorem which applies to the solution to SPDE (5) under some conditions on the fundamental solution associated to the operator $\mathcal{L}$. The last section focuses on a further investigation of the solution to SPDE (7). In which case, we show that the ASCLT is statisfied.
2 Existence and some characteristics of the solution
The notion of solution to (5) is defined in the mild sense. We call a mild solution to (5) the stochastic process
where W is the Gaussian noise with covariance given by (53), G is the fundamental solution of the operator $\mathcal{L}$ and the integral in (8) is the Wiener integral with respect to the Gaussian noise W. The existence and many properties of the fundamental solution G of the operator $\mathcal{L}$ have been obtained in many papers (see, e.g., [13] and [19]).
(8)
\[ u(t,x)={\int _{0}^{t}}{\int _{\mathbb{R}}}G(t-s,x,y)W(ds,dy),\hspace{1em}t\in [0,T],x\in \mathbb{R},\]Remark 1.
It is well known (see, e.g., [25]) that the mild solution to (5) exists when the Wiener integral (8) is well-defined and this happens when the function $(s,y)\longmapsto G(t-s,x,y)$ belongs to ${\mathcal{H}_{0}}={L^{2}}([0,T]\times \mathbb{R})$, the canonical Hilbert space associated with the Gaussian process W. In fact, ${\mathcal{H}_{0}}$ is none other than the closure of the linear span generated by the indicator functions ${\mathbf{1}_{[0,t]\times C}}$, $t\in [0,T]$, $C\in {\mathcal{B}_{b}}(\mathbb{R})$, with respect to the inner product
Moreover, the process $(u(t,x),t\in [0,T],x\in \mathbb{R})$, when it exists, is a centered Gaussian process.
The following proposition deals with the existence of the mild solution to Equation (5).
Proof.
The existence and some bounds of the fundamental solution to PDE (1) have been established in [1] and [9]. In particular, it has been proved that there exist positive constants ${C_{1}}$ and ${C_{2}}$ such that
for any $t\in [0,T]$ and $(x,y)\in {\mathbb{R}^{2}}$. Thus,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}{G^{2}}(t-s,x,y)dyds& \displaystyle \le & \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}\frac{{C_{1}^{2}}}{2\pi (t-s)}\exp \bigg(-\frac{2{C_{2}}{(x-y)^{2}}}{t-s}\bigg)dyds\\ {} & \displaystyle \le & \displaystyle {C_{3}}{\int _{0}^{t}}\frac{1}{\sqrt{t-s}}ds\\ {} & \displaystyle \le & \displaystyle {C_{4}}\sqrt{T},\end{array}\]
where ${C_{3}}$ and ${C_{4}}$ denote two strictly positive constants. This with Remark 1 and Wiener’s isometry allow us to get the existence of the mild solution to (5) and to show that
\[ \mathbb{E}\big(u{(t,x)^{2}}\big)={\int _{0}^{t}}{\int _{\mathbb{R}}}{G^{2}}(t-s,x,y)dyds\le {C_{4}}\sqrt{T},\]
for every $t\in [0,T]$ and $x\in \mathbb{R}$. □Now we consider an interval I in $\mathbb{R}$ and denote
and
\[ {\big\| {\Delta _{h}}G(t-.,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}={\int _{0}^{t}}{\int _{\mathbb{R}}}{\big({\Delta _{h}}G(t-\sigma ,x,y)\big)^{2}}\hspace{0.1667em}d\sigma \hspace{0.1667em}dy\]
for every $u,t\in (0,T]$, $h>0$ and $x,z\in I$. We also consider the conditions:The following lemma will play an important role in this paper.
Proof.
We first note that if $x=y$, then Inequalities (10) and (11) are trivial. We also note that the proofs in the cases $x>y$ and $y>x$ are similar. So we consider only the case $y>x$. Using Wiener’s isometry we get
Equality (13) and Condition ${H_{1}}(I)$ [respectively ${H_{2}}(I)$] allow us to get the two first assertions in Lemma 1.
(13)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \mathbb{E}{\big(u(t,y)-u(t,x)\big)^{2}}\\ {} & \displaystyle =& \displaystyle \mathbb{E}{\bigg({\int _{(0,t)\times \mathbb{R}}}G(t-u,y,z)W(du,dz)\hspace{0.1667em}-{\int _{(0,t)\times \mathbb{R}}}G(t-u,x,z)W(du,dz)\bigg)^{2}}\\ {} & \displaystyle =& \displaystyle \mathbb{E}{\bigg({\int _{(0,t)\times \mathbb{R}}}\big(G(t-u,y,z)-G(t-u,x,z)\big)W(du,dz)\bigg)^{2}}\\ {} & \displaystyle =& \displaystyle {\int _{(0,t)\times \mathbb{R}}}{\big(G(t-u,y,z)-G(t-u,x,z)\big)^{2}}du\hspace{0.1667em}dz\\ {} & \displaystyle =& \displaystyle {\big\| {\Delta _{y-x}}G(t-.,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}.\end{array}\]As for the third one, using again Wiener’s isometry we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \mathbb{E}\big(\big(u(t,x+h)-u(t,x)\big)\big(u(t,y+h)-u(t,y)\big)\big)\\ {} & \displaystyle =& \displaystyle \mathbb{E}\bigg({\int _{(0,t)\times \mathbb{R}}}{\Delta _{h}}G(t-u,x,z)W(du,dz)\\ {} & & \displaystyle \times {\int _{(0,t)\times \mathbb{R}}}{\Delta _{h}}G(t-u,y,z)W(du,dz)\bigg)\\ {} & \displaystyle =& \displaystyle {\int _{(0,t)\times \mathbb{R}}}{\Delta _{h}}G(t-u,x,z)\hspace{0.1667em}{\Delta _{h}}G(t-u,y,z)\hspace{0.2778em}du\hspace{0.1667em}dz.\end{array}\]
Using Condition ${H_{3}}(I)$ the proof of the third assertion in Lemma 1 is achieved. □From Assertion 2 in Lemma 1 and by Kolmogorov’s criterion of continuity, we easily get the following corollary.
Corollary 1.
Let u be the mild solution to (5). If Condition ${H_{2}}(I)$ is satisfied, then, for every $t\in [0,T]$, the process ${(u(t,x))_{x\in I}}$ is Hölder continuous of order γ with $0<\gamma <\frac{1}{2}$.
3 Almost sure central limit theorem
Let us start this section with the following definition.
Definition 1.
Let ${({G_{N}})}_{N\ge 1}$ be a sequence of real-valued random variables defined on a common probability space $(\varOmega ,\mathcal{F},\mathbb{P})$. We say that the sequence ${({G_{N}})}_{N\ge 1}$ satisfies an almost sure central limit theorem (ASCLT), if, almost surely, for every bounded and continuous function $\varphi :\mathbb{R}\to \mathbb{R}$, we have:
\[ \frac{1}{logN}{\sum \limits_{i=1}^{N}}\frac{\varphi ({G_{i}})}{i}\longrightarrow \mathbb{E}\big(\varphi (\mathcal{Z})\big)\hspace{1em}\text{as}\hspace{2.5pt}N\longrightarrow \infty ,\]
where $\mathcal{Z}$ is an $\mathcal{N}(0,1)$ random variable.For fixed $t\in (0,T]$, we consider the Gaussian process ${(u(t,x))_{x\in [0,1]}}$ being the mild solution to Equation (5). We also consider the partition $0={x_{0}}<{x_{1}}<\cdots <{x_{N}}=1$ of the interval $[0,1]$ defined by ${x_{i}}=\frac{i}{N}$ for every $i=0,1,\dots ,N$. We define the centered re-normalized quadratic variation statistic in the following way:
The aim of this section is to show that the sequence ${({\tilde{V}_{N}})}_{N\ge 1}$ satisfies the ASCLT. Let us first recall briefly some basic elements of the Stein–Malliavin theory (see [16]) that will be useful in our proof.
3.1 Elements of the Stein–Malliavin theory
Consider a real separable Hilbert space $(\mathcal{H},<.,.{>_{\mathcal{H}}})$ and an isonormal Gaussian process $(B(\varphi ),\varphi \in \mathcal{H})$ on a probability space $(\varOmega ,\mathcal{F},\mathbb{P})$, which is a centered Gaussian family of random variables such that
for every $\varphi ,\psi \in \mathcal{H}$. For $q\ge 1$, let ${\mathcal{H}^{\otimes q}}$ be the qth tensor product of $\mathcal{H}$ and denote ${\mathcal{H}^{\odot q}}$ the associated qth symmetric tensor product.
Denote by ${I_{q}}$ the qth multiple stochastic integral with respect to B. This ${I_{q}}$ is actually an isometry between the Hilbert space ${\mathcal{H}^{\odot q}}$ equipped with the scaled norm $\frac{1}{\sqrt{q!}}\hspace{0.2778em}\| .{\| _{{\mathcal{H}^{\otimes q}}}}$ and the Wiener chaos of order q, which is defined as the closed linear span of the random variables ${H_{q}}(B(\varphi ))$, where $\varphi \in \mathcal{H},\hspace{0.2778em}\hspace{0.2778em}\| \varphi {\| _{\mathcal{H}}}=1$ and ${H_{q}}$ is the Hermite polynomial of degree $q\geqslant 1$ defined by
The isometry of multiple integrals can be written as follows: for $p,\hspace{0.1667em}q\geqslant 1,f\in {\mathcal{H}^{\otimes p}}$ and $g\in {\mathcal{H}^{\otimes q}}$,
It holds that
where $\hat{f}$ denotes the canonical symmetrization of f defined by
where the sum runs over all permutations σ of $\{1,\dots ,q\}$.
(15)
\[ {H_{q}}(x)=\frac{{(-1)^{q}}}{q!}\hspace{0.1667em}\exp \bigg(\frac{{x^{2}}}{2}\bigg)\hspace{0.1667em}\frac{{d^{q}}}{d{x^{q}}}\bigg(\exp \bigg(-\frac{{x^{2}}}{2}\bigg)\bigg),\hspace{1em}x\in \mathbb{R}.\](16)
\[ \mathbb{E}\big({I_{p}}(f){I_{q}}(g)\big)=\left\{\begin{array}{l@{\hskip10.0pt}l}q!<\hat{f},\hat{g}{>_{{\mathcal{H}^{\otimes q}}}}\hspace{1em}& \text{if}\hspace{0.1667em}p=q\\ {} 0\hspace{1em}& \text{otherwise}.\end{array}\right.\](17)
\[ \hat{f}({x_{1}},\dots ,{x_{q}})=\frac{1}{q!}\sum \limits_{\sigma \in {S_{q}}}f({x_{\sigma (1)}},\dots ,{x_{\sigma (q)}}),\]We recall that any square-integrable random variable F, which is measurable with respect to the σ-algebra generated by B, can be expanded into an orthogonal sum of multiple stochastic integrals:
where the series converges in the ${L^{2}}(\varOmega )$-sense and the kernels ${f_{q}}$, belonging to ${\mathcal{H}^{\odot q}}$, are uniquely determined by F.
Consider now the class of smooth random variables F that can be written in the form
where $n\geqslant 1$, $g:{\mathbb{R}^{n}}\longmapsto \mathbb{R}$ is a ${\mathcal{C}^{\infty }}$-function with compact support and ${\varphi _{1}},\dots ,{\varphi _{n}}\in \mathcal{H}$. The Malliavin derivative of a smooth random variable F of the form (19) is the $\mathcal{H}$-valued random variable given by
The following formula for multiplication of Wiener chaos integrals of any orders $p,q$ will play a basic role in the next section. For any symmetric integrands $f\in {\mathcal{H}^{\odot p}}$ and $g\in {\mathcal{H}^{\odot p}}$, we have
where, in the particular case when $\mathcal{H}={L^{2}}([0,T])$, for $r=1,\dots ,p\wedge q$, the rth contraction $f{\otimes _{r}}g$ is the element of ${\mathcal{H}^{\otimes (p+q-2r)}}$ defined by
(21)
\[ {I_{p}}(f){I_{q}}(g)={\sum \limits_{r=0}^{p\wedge q}}r!\big({_{r}^{p}}\big)\big({_{r}^{q}}\big){I_{p+q-2r}}(f{\otimes _{r}}g),\](22)
\[ \begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & (f{\otimes _{r}}g)({s_{1}},\dots ,{s_{p-r}},{t_{1}},\dots ,{t_{q-r}})\\ {} & =& {\displaystyle \int _{{[0,T]^{r}}}}d{u_{1}}...d{u_{r}}f({s_{1}},\dots ,{s_{p-r}},{u_{1}},\dots ,{u_{r}})\hspace{0.1667em}g({t_{1}},\dots ,{t_{q-r}},{u_{1}},\dots ,{u_{r}}).\end{array}\]The following theorem gives a description of the normal approximation of multiple stochastic integrals. We refer to [16–18] and references therein for the proof.
Theorem 1.
Fix $q\geqslant 1$. Assume that ${({G_{N}})}_{N\geqslant 1}:={({\mathrm{I}_{q}}({g_{N}}))}_{N\geqslant 1}$ with ${g_{N}}\in {\mathcal{H}^{\odot q}}$ is a sequence of random variables belonging to the qth Wiener chaos such that
Hence, ${G_{N}}$ converges in law to $\mathcal{Z}\sim \mathcal{N}(0,1)$ if and only if
Furthermore, if we denote by d one of the metrics on the space of probability measures on $\mathbb{R}$, including the Kolmogorov, Wasserstein and Total Variation measures, then for N large enough:
The following theorem has been introduced in [4]. It gives a sufficient condition for extending Theorem 1 to an ASCLT for multiple stochastic integrals.
Theorem 2.
Fix $q\geqslant 2$, and let ${({G_{N}})}_{N\geqslant 1}$ be a sequence of random variables defined by
Then, the sequence ${({G_{N}})}_{N\ge 1}$ satisfies an ASCLT.
\[ {G_{N}}:={\big({\mathrm{I}_{q}}({g_{N}})\big)}_{N\geqslant 1};\hspace{1em}{g_{N}}\in {\mathcal{H}^{\odot q}}.\]
Suppose that:
-
1. For every $N\geqslant 1,\hspace{0.2778em}\mathbb{E}({G_{N}^{2}})=1$.
-
2. For every $r=1,\dots ,q-1$, $\underset{N\to \infty }{\lim }\| {g_{N}}{\otimes _{r}}{g_{N}}{\| _{{\mathcal{H}^{\otimes 2(q-r)}}}^{2}}=0$.
-
3. For every $r=1,\dots ,q-1,\hspace{0.2778em}$ $\displaystyle\sum \limits_{N\geqslant 2}\displaystyle\frac{1}{Nlo{g^{2}}N}\hspace{0.1667em}{\displaystyle\sum \limits_{l=1}^{N}}\displaystyle\frac{1}{l}\| {g_{l}}{\otimes _{r}}{g_{l}}{\| _{{\mathcal{H}^{\otimes 2(q-r)}}}^{2}}<\infty $.
-
4. $\displaystyle\sum \limits_{N\geqslant 2}\displaystyle\frac{1}{Nlo{g^{3}}N}\hspace{0.1667em}{\displaystyle\sum \limits_{i,j=1}^{N}}\displaystyle\frac{|\mathbb{E}({G_{i}}{G_{j}})|}{ij}<\infty $.
We finish this section with the following useful reduction lemma. For its proof see Lemma 2.2 in [2].
3.2 Limiting behavior of the re-normalized quadratic variation of the spatial solution process
For fixed $t\in (0,T]$, we denote by $\mathcal{H}$ the canonical Hilbert space associated to the Gaussian process ${(u(t,x))_{x\in [0,1]}}$ being a mild solution to Equation (5). This Hilbert space is defined as the closure of the linear span generated by the indicator functions ${\mathbf{1}_{[0,x]}},\hspace{0.2778em}x>0$, with respect to the inner product
We also denote by ${I_{q}}$, $q\ge 1$, the multiple stochastic integral with respect to the Gaussian process ${(u(t,x))_{x\in [0,1]}}$. So for every $x<y$ we have
(23)
\[ \mathbb{E}\big(u(t,x)u(t,y)\big)=<{\mathbf{1}_{[0,x]}},{\mathbf{1}_{[0,y]}}{>_{\mathcal{H}}}.\]We start our study of the limit behavior in distribution of the sequence ${({\tilde{V}_{N}})}_{N\ge 1}$ by the following main theorem.
Theorem 3.
Proof.
By using Formula (21), we can write
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {V_{N}}& \displaystyle =& \displaystyle {\sum \limits_{j=0}^{N-1}}\bigg[\frac{{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}}-1\bigg]\\ {} & \displaystyle =& \displaystyle {\sum \limits_{j=0}^{N-1}}\bigg[\frac{{{\mathrm{I}_{1}}^{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}})}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}}-1\bigg]\\ {} & \displaystyle =& \displaystyle {\sum \limits_{j=0}^{N-1}}\frac{{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}^{{\otimes ^{2}}}})}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}}.\end{array}\]
By virtue of the isometry formula (16), we get
and
On the one hand we clearly have ${T_{1,N}}=2N$. On the other hand, since Conditions ${H_{1}}([0,1])$ and ${H_{3}}([0,1])$ are satisfied, by virtue of Lemma 1 we get
where C denotes a universal positive constant. Thus, we deduce that the dominant term for $\mathbb{E}({\tilde{V}_{N}^{2}})$ is obviously ${T_{1,N}}$. Consequently, we obtain, for a fixed $t\in (0,T]$,
□
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \mathbb{E}\big({V_{N}^{2}}\big)& \displaystyle =& \displaystyle \mathbb{E}{\Bigg({\sum \limits_{j=0}^{N-1}}\frac{{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}^{{\otimes ^{2}}}})}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}}\Bigg)^{2}}\\ {} & \displaystyle =& \displaystyle {\sum \limits_{j,k=0}^{N-1}}\frac{\mathbb{E}({\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}^{{\otimes ^{2}}}})\hspace{0.1667em}{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{k}},{x_{k+1}}]}^{{\otimes ^{2}}}}))}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & \displaystyle =& \displaystyle 2{\sum \limits_{j,k=0}^{N-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}^{2}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}.\end{array}\]
Thus,
where
(24)
\[ {T_{1,N}}=2{\sum \limits_{j=0}^{N-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}{>_{\mathcal{H}}^{2}}}{{[\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}]^{2}}}\](25)
\[ {T_{2,N}}=2{\sum \limits_{j,k=0;j\ne k}^{N-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}^{2}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}.\](26)
\[ {T_{2,N}}\leqslant 2\hspace{0.1667em}{N^{2}}{\sum \limits_{j,k=0;j\ne k}^{N-1}}<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}^{2}}\leqslant C\hspace{0.1667em}{N^{2}}\hspace{0.1667em}{\sum \limits_{j,k=0;j\ne k}^{N-1}}{\bigg(\frac{1}{{N^{2}}}\bigg)^{2}}\le C,\]In the following theorem we establish the convergence in law of the sequence ${({\tilde{V}_{N}})}_{N}$.
Theorem 4.
Consider the sequence of random variables ${\tilde{V}_{N}}$ defined in (14). If G satisfies Conditions ${H_{1}}([0,1])$ and ${H_{3}}([0,1])$, then
\[ {\tilde{V}_{N}}\hspace{0.2778em}\stackrel{Law}{\longrightarrow }\hspace{0.2778em}\mathcal{N}(0,1).\]
Moreover, if we denote by d one of the metrics on the space of probability measures on $\mathbb{R}$, including the Kolmogorov, Wasserstein and Total Variation measures, then for N large enough
Proof.
By virtue of Formula (20) we get
\[ D{\tilde{V}_{N}}=\frac{1}{\sqrt{2N}\hspace{0.1667em}}{\sum \limits_{j=0}^{N-1}}\frac{{\mathrm{I}_{1}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}})\hspace{0.2778em}{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}}.\]
Hence, for every fixed $t\in [0,T]$, using Formula (21), we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}& \displaystyle =& \displaystyle \frac{2}{N}{\sum \limits_{j,k=0}^{N-1}}\frac{{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\otimes {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}})\hspace{0.1667em}<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle +\mathbb{E}\big(\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}\big),\end{array}\]
and consequently,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \mathbf{Var}\big(\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}\big)\\ {} & \displaystyle =& \displaystyle \mathbb{E}{\big[\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}-\mathbb{E}\big(\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}\big)\big]^{2}}\\ {} & \displaystyle =& \displaystyle \mathbb{E}{\Bigg[\frac{2}{N}{\sum \limits_{j,k=0}^{N-1}}\frac{{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\otimes {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}})\hspace{0.1667em}<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\Bigg]^{2}}\\ {} & \displaystyle =& \displaystyle \frac{8}{{N^{2}}}{\sum \limits_{j,k,m,l=0}^{N-1}}\frac{\mathbb{E}({\mathrm{I}_{2}}({\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\otimes {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}})\hspace{0.1667em}{\mathrm{I}_{2}}({\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}\otimes {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}))}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{m+1}})-u(t,{x_{m}}))^{2}}\hspace{0.2778em}\mathbb{E}{(u(t,{x_{l+1}})-u(t,{x_{l}}))^{2}}}\\ {} & \displaystyle =& \displaystyle \frac{8}{{N^{2}}}{\sum \limits_{j,k,m,l=0}^{N-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}}{\| {\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\| {\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{2}}},\end{array}\]
where $f\tilde{\otimes }g$ denotes the symmetrization of the tensor product $f\otimes g$ that satisfies
and
\[ <f\tilde{\otimes }g,{f^{\prime }}\tilde{\otimes }{g^{\prime }}{>_{\mathcal{H}}}=\frac{1}{2}\big(<f,{f^{\prime }}{>_{\mathcal{H}}}<g,{g^{\prime }}{>_{\mathcal{H}}}+<f,{g^{\prime }}{>_{\mathcal{H}}}<g,{f^{\prime }}{>_{\mathcal{H}}}\big).\]
Therefore,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \mathbf{Var}\big(\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}\big)\\ {} & \displaystyle =& \displaystyle \frac{8}{{N^{2}}}{\sum \limits_{j,k,m,l=0}^{N-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\| {\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{>_{\mathcal{H}}}}{\| {\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{\| _{\mathcal{H}}^{2}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & \displaystyle =& \displaystyle {D_{4,N}}+{D_{3,N}}+{D_{2,N}}+{D_{1,N}}\end{array}\]
where ${D_{i,N}}$, for every $i\in \{1,2,3,4\}$, contains all the terms with i equal indices. So, ${D_{4,N}}$ contains all the summands above with $j=k=m=l$; that is
As for ${D_{3,N}}$, it contains all the terms corresponding to $j=k=l\ne m$; so, since G satisfies Conditions ${H_{1}}([0,1])$ and ${H_{3}}([0,1])$, using Lemma 1 we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {D_{3,N}}& \displaystyle \le & \displaystyle \frac{8}{{N^{2}}}{\sum \limits_{l,m=0}^{N-1}}\frac{\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{4}}<{\mathbf{1}_{[{x_{l}},{x_{l+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{>_{\mathcal{H}}^{2}}}{\| {\mathbf{1}_{[{x_{l}},{x_{l+1}}]}}{\| _{\mathcal{H}}^{6}}\hspace{0.2778em}\| {\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{\| _{\mathcal{H}}^{2}}}\\ {} & \displaystyle \le & \displaystyle \frac{C}{{N^{2}}}{\sum \limits_{l,m=0}^{N-1}}\frac{{(\frac{1}{{N^{2}}})^{2}}}{{(\frac{1}{N})^{2}}}=\frac{C}{{N^{2}}}.\end{array}\]
By the same way, and using again Lemma 1, we show that
\[ {D_{2,N}}\leqslant \frac{C}{{N^{2}}}\hspace{1em}\text{and}\hspace{1em}{D_{1,N}}\leqslant \frac{C}{{N^{2}}}.\]
All this allow us to get
Moreover, we have
\[ \mathbb{E}\big(\| D{\tilde{V}_{N}}{\| _{\mathcal{H}}^{2}}\big)=2\mathbb{E}{({\tilde{V}_{N}})^{2}}=\frac{\mathbb{E}({V_{N}^{2}})}{N}=\frac{1}{N}({T_{1,N}}+{T_{2,N}})=2+\frac{{T_{2,N}}}{N}\]
where ${T_{1,N}}$ and ${T_{2,N}}$ are defined by (24) and (25). This and Inequality (26) allow us to deduce that
By virtue of Theorem 1, the proof of Theorem 4 is completed. □3.3 Almost sure central limit theorem
The following theorem is a kind of extension of Theorem 4.
Theorem 5.
If G satisfies Conditions ${H_{1}}([0,1])$ and ${H_{3}}([0,1])$, then the sequence ${({\tilde{V}_{N}})}_{N\ge 1}$ satisfies an ASCLT.
Proof.
Denoting ${\sigma _{N}}=\sqrt{\mathbb{E}({\tilde{V}_{N}^{2}})}$, for every $N\ge 1$, according to Theorem 3, we have ${\lim \nolimits_{N\to \infty }}{\sigma _{N}}=1$. So without loss of generality, we assume that ${\inf _{N\ge 1}}{\sigma _{N}}={\sigma _{0}}>0$ and we consider ${G_{N}}=\frac{{\tilde{V}_{N}}}{{\sigma _{N}}}$, for every $N\ge 1$.
According to Lemma 2, to obtain Theorem 5 it suffices to show that the sequence ${({G_{N}})}_{N\ge 1}$ satisfies an ASCLT. To this end, since for every $N\ge 1$ we have ${G_{N}}={\mathrm{I}_{2}}({g_{N}})$ with
\[ {g_{N}}:=\frac{1}{{\sigma _{N}}\sqrt{2N}}{\sum \limits_{j=0}^{N-1}}\frac{{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}^{{\otimes ^{2}}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}},\]
and since we obviously have $\mathbb{E}({G_{N}^{2}})=1$, for every $N\ge 1$, it suffices to check the three last assumptions in Theorem 2.By the 1st contraction defined by (22), we obtain
and consequently the second assumption in Theorem 2 is satisfied.
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {g_{l}}{\otimes _{1}}{g_{l}}& \displaystyle =& \displaystyle \frac{1}{2{\sigma _{l}^{2}}l}{\sum \limits_{j,k=0}^{l-1}}\frac{{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}^{{\otimes ^{2}}}}{\otimes _{1}}{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}^{{\otimes ^{2}}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & \displaystyle =& \displaystyle \frac{1}{2{\sigma _{l}^{2}}l}{\sum \limits_{j,k=0}^{l-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\otimes {\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}.\end{array}\]
Therefore,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \| {g_{l}}{\otimes _{1}}{g_{l}}{\| _{{\mathcal{H}^{{\otimes ^{2}}}}}^{2}}\\ {} & \displaystyle =& \displaystyle \frac{1}{4{\sigma _{l}^{4}}{l^{2}}}{\sum \limits_{j,k,m,p=0}^{l-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}\tilde{\otimes }{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{m+1}})-u(t,{x_{m}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{p+1}})-u(t,{x_{p}}))^{2}}}\\ {} & \displaystyle =& \displaystyle \frac{1}{4{\sigma _{l}^{2}}{l^{2}}}{\sum \limits_{j,k,m,p=0}^{l-1}}\frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}},{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{j+1}})-u(t,{x_{j}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{k+1}})-u(t,{x_{k}}))^{2}}}\\ {} & & \displaystyle \times \frac{<{\mathbf{1}_{[{x_{j}},{x_{j+1}}]}},{\mathbf{1}_{[{x_{m}},{x_{m+1}}]}}{>_{\mathcal{H}}}\hspace{0.1667em}<{\mathbf{1}_{[{x_{k}},{x_{k+1}}]}},{\mathbf{1}_{[{x_{p}},{x_{p+1}}]}}{>_{\mathcal{H}}}}{\mathbb{E}{(u(t,{x_{m+1}})-u(t,{x_{m}}))^{2}}\hspace{0.1667em}\mathbb{E}{(u(t,{x_{p+1}})-u(t,{x_{p}}))^{2}}}.\end{array}\]
Since $\frac{1}{{\sigma _{l}^{4}}}\le \frac{1}{{\sigma _{0}^{4}}}$ for every $l\ge 1$, and since G satisfies Conditions ${H_{1}}([0,1])$ and ${H_{3}}([0,1])$, proceeding in the same way as in the proof of Theorem 4, we get
(27)
\[ \| {g_{l}}{\otimes _{1}}{g_{l}}{\| _{{\mathcal{H}^{{\otimes ^{2}}}}}^{2}}\leqslant \frac{C}{l},\]From Inequality (27) we also deduce that
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \sum \limits_{N\geqslant 2}\frac{1}{Nlo{g^{2}}N}\hspace{0.1667em}{\sum \limits_{l=1}^{N}}\frac{1}{l}\| {g_{l}}{\otimes _{1}}{g_{l}}{\| _{{\mathcal{H}^{\otimes 2}}}^{2}}& \displaystyle \leqslant & \displaystyle C\hspace{0.1667em}\sum \limits_{N\geqslant 2}\frac{1}{Nlo{g^{2}}N}\hspace{0.1667em}{\sum \limits_{l=1}^{\infty }}\frac{1}{{l^{2}}}\\ {} & \displaystyle \leqslant & \displaystyle C\hspace{0.1667em}\sum \limits_{N\geqslant 2}\frac{1}{Nlo{g^{2}}N}<\infty ,\end{array}\]
that means that the third assumption in Theorem 2 is also satisfied.Let us now check the last assumption in Theorem 2. Since we have
and since Conditions ${H_{1}}([0,1])$ and ${H_{3}}([0,1])$ are satisfied, using Lemma 1 we get:
\[ \text{If}\hspace{2.5pt}i=j,\hspace{1em}<{g_{i}},{g_{i}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}\leqslant \frac{C}{i}\hspace{1em}\text{and}\hspace{1em}\text{if}\hspace{2.5pt}i>j,\hspace{1em}<{g_{i}},{g_{j}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}\leqslant C\hspace{0.1667em}\sqrt{\frac{j}{i}}.\]
Therefore,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \sum \limits_{N\geqslant 2}\frac{1}{Nlo{g^{3}}N}\hspace{0.1667em}{\sum \limits_{i,j=1}^{N}}\frac{|\mathbb{E}({G_{i}}{G_{j}})|}{ij}\\ {} & \displaystyle =& \displaystyle \sum \limits_{N\geqslant 2}\frac{1}{Nlo{g^{3}}N}\hspace{0.1667em}\Bigg[{\sum \limits_{i\ne j=1}^{N}}\frac{|\mathbb{E}({G_{i}}{G_{j}})|}{ij}+{\sum \limits_{i=1}^{N}}\frac{|\mathbb{E}({G_{i}^{2}})|}{{i^{2}}}\Bigg]\\ {} & \displaystyle \leqslant & \displaystyle 2\sum \limits_{N\geqslant 2}\frac{1}{Nlo{g^{3}}N}\hspace{0.1667em}\Bigg[2{\sum \limits_{i>j=1}^{N}}\frac{|<{g_{i}},{g_{j}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}|}{ij}+{\sum \limits_{i=1}^{N}}\frac{|<{g_{i}},{g_{i}}{>_{{\mathcal{H}^{{\otimes ^{2}}}}}}|}{{i^{2}}}\Bigg]\\ {} & \displaystyle \leqslant & \displaystyle C\hspace{0.1667em}\sum \limits_{N\geqslant 2}\frac{2}{Nlo{g^{3}}N}\hspace{0.1667em}\Bigg[{\sum \limits_{i>j=1}^{N}}\frac{2}{i\sqrt{ij}}+{\sum \limits_{i=1}^{N}}\frac{1}{{i^{3}}}\Bigg]\\ {} & \displaystyle <& \displaystyle \infty .\hspace{280.0pt}\end{array}\]
□4 Stochastic heat equation with piecewise constant coefficients
The study done in the previous section allows us to make a new step in the investigation of the solution to the SPDE (7). Equation (7) is obviously a particular case of (5). Indeed, the operator ${\mathcal{L}_{p}}$ defined by (3) can be written in the form (2) with
In the following proposition we present the expression of the fundamental solution associated to the operator ${\mathcal{L}_{p}}$. For a proof see, e.g., [8, 26, 27] and [28].
Proposition 2.
There exists a unique fundamental solution $G(t-s,x,y)$ associated to the operator ${\mathcal{L}_{p}}$. It can be explicitly expressed as
with
(28)
\[ G(u,x,z)=m(u)\hspace{0.2778em}\bigg[\frac{1}{\sqrt{{a_{1}}}}\hspace{0.1667em}{A^{-}}(u,x,z){1_{\{z\leqslant 0\}}}+\frac{1}{\sqrt{{a_{2}}}}\hspace{0.1667em}{A^{+}}(u,x,z){1_{\{z>0\}}}\bigg]\](30)
\[ \left\{\begin{array}{l}{A^{-}}(u,x,z)={E^{-}}(u,x,z)-\beta {E^{+}}(u,x,z),\\ {} {A^{+}}(u,x,z)={E^{-}}(u,x,z)+\beta {E^{+}}(u,x,z),\end{array}\right.\]In this section, by making an in-depth study of the terms $f,{A^{-}}$ and ${A^{+}}$ defined in Expressions (30) and (32), we will prove the following theorem.
Theorem 6.
Let u be the mild solution to Equation (7) and ${\tilde{V}_{N}}$ be the sequence given by (14). Suppose that the coefficients A and ρ defined in (4) satisfy
Then the following is valid:
(33)
\[ \max \bigg(1,\frac{\sqrt{{a_{1}}}}{\sqrt{{a_{2}}}}\bigg)\le \frac{{\rho _{2}}}{{\rho _{1}}}.\]To prove Theorem 6, we shall first establish the following lemmas.
4.1 Preliminary lemmas
Proof.
Expression (32) allows to get
If $xy\ge 0$, both inequalities in Lemma 3 are directly obtained from (34). If $y>0$ and $x<0$,
(34)
\[ f(y)-f(x)=\left\{\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \frac{y-x}{\sqrt{{a_{2}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y>0\hspace{0.2778em}x>0,\\ {} \displaystyle \frac{y-x}{\sqrt{{a_{1}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y\le 0\hspace{0.2778em}x\le 0,\\ {} \displaystyle \frac{y}{\sqrt{{a_{1}}}}-\displaystyle \frac{x}{\sqrt{{a_{2}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y\le 0\hspace{0.2778em}x>0,\\ {} \displaystyle \frac{y}{\sqrt{{a_{2}}}}-\displaystyle \frac{x}{\sqrt{{a_{1}}}}\hspace{1em}& \text{if}\hspace{2.5pt}& y>0\hspace{0.2778em}x\le 0.\end{array}\right.\]
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \max \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)|y-x|-\big|f(y)-f(x)\big|\\ {} & \displaystyle =& \displaystyle \max \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)(y-x)-\frac{y}{\sqrt{{a_{2}}}}+\frac{x}{\sqrt{{a_{1}}}}\\ {} & \displaystyle =& \displaystyle y\bigg[\max \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)-\frac{1}{\sqrt{{a_{2}}}}\bigg]-x\bigg[\max \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)-\frac{1}{\sqrt{{a_{1}}}}\bigg]\\ {} & \displaystyle >& \displaystyle 0\end{array}\]
and
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \min \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)|y-x|-\big|f(y)-f(x)\big|\\ {} & \displaystyle =& \displaystyle \min \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)(y-x)-\frac{y}{\sqrt{{a_{2}}}}+\frac{x}{\sqrt{{a_{1}}}}\\ {} & \displaystyle =& \displaystyle y\bigg[\min \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)-\frac{1}{\sqrt{{a_{2}}}}\bigg]-x\bigg[\min \bigg(\frac{1}{\sqrt{{a_{2}}}},\frac{1}{\sqrt{{a_{1}}}}\bigg)-\frac{1}{\sqrt{{a_{1}}}}\bigg]\\ {} & \displaystyle <& \displaystyle 0.\end{array}\]
The proof of both inequalities in the case where $y<0$ and $x>0$ is similar. □Lemma 4.
There exists a universal positive constant C, such that
\[ {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{+}}(u,y,z)-{E^{+}}(u,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\leqslant C\hspace{0.2778em}|y-x|\]
and
\[ {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{-}}(u,y,z)-{E^{-}}(u,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\leqslant C\hspace{0.2778em}|y-x|\]
for every $t>0$ and $x,y\in \mathbb{R}$.
Proof.
We present only the proof of the first inequality; the proof of the second one is similar. By using Expression (31), we get
Therefore,
where $\tilde{H}=|f(y)|-|f(x)|$ and ${p_{u}}$ denotes the heat kernel defined by
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{+}}(u,y,z)-{E^{+}}(u,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(|f(z)|+|f(y)|)^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(|f(z)|+|f(x)|)^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}{\int _{0}^{\infty }}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(\frac{{(z/\sqrt{{a_{2}}}+|f(y)|)^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(z/\sqrt{{a_{2}}}+|f(x)|)^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & & \displaystyle +{\int _{0}^{t}}{\int _{-\infty }^{0}}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(-z/\sqrt{{a_{1}}}+|f(y)|)^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(-z/\sqrt{{a_{1}}}+|f(x)|)^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du.\end{array}\]
The changes of variables $Z=z/\sqrt{{a_{2}}}+|f(x)|$ in the first integral and $Z=-z/\sqrt{{a_{1}}}+|f(x)|$, in the second one give
(35)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{+}}(u,y,z)-{E^{+}}(u,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle \sqrt{{a_{2}}}\hspace{0.2778em}{\int _{0}^{t}}{\int _{|f(x)|}^{+\infty }}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(Z+(|f(y)|-|f(x)|))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{Z^{2}}}{2u}\bigg)\hspace{0.2778em}\bigg]{^{2}}\hspace{0.1667em}dZ\hspace{0.2778em}du\\ {} & & \displaystyle +\sqrt{{a_{1}}}\hspace{0.2778em}{\int _{0}^{t}}{\int _{|f(x)|}^{+\infty }}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(Z+(|f(y)|-|f(x)|))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{Z^{2}}}{2u}\bigg)\hspace{0.2778em}\bigg]{^{2}}\hspace{0.1667em}dZ\hspace{0.2778em}du.\end{array}\](36)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{+}}(u,y,z)-{E^{+}}(u,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle \leqslant & \displaystyle 2\max (\sqrt{{a_{1}}},\sqrt{{a_{2}}})\hspace{0.2778em}{\int _{0}^{t}}{\int _{\mathbb{R}}}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(Z+\tilde{H})^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{Z^{2}}}{2u}\bigg)\hspace{0.2778em}\bigg]{^{2}}\hspace{0.1667em}dZ\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle 2\max (\sqrt{{a_{1}}},\sqrt{{a_{2}}})\hspace{0.2778em}{\int _{0}^{t}}{\int _{\mathbb{R}}}{\big[{p_{u}}(Z+\tilde{H})-{p_{u}}(Z)\hspace{0.2778em}\big]^{2}}\hspace{0.1667em}dZ\hspace{0.2778em}du,\end{array}\]It is known that the Fourier transform of ${p_{u}}$ is
\[ \mathcal{F}({p_{t}})(\xi )={e^{-t{\xi ^{2}}/2}}\hspace{1em}\forall \hspace{0.1667em}\xi \in \mathbb{R},\hspace{0.1667em}t>0.\]
By virtue of the Plancherel theorem we can write
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}ds{\int _{\mathbb{R}}}{\big[{p_{s}}(v+h)-{p_{s}}(v)\big]^{2}}dv\\ {} & \displaystyle =& \displaystyle \frac{1}{2\pi }{\int _{0}^{t}}ds{\int _{-\infty }^{\infty }}{\big|{e^{-s{\xi ^{2}}/2+i\xi h}}-{e^{-s{\xi ^{2}}/2}}\big|^{2}}\hspace{0.2778em}d\xi \\ {} & \displaystyle =& \displaystyle \frac{1}{\pi }{\int _{0}^{t}}ds{\int _{-\infty }^{\infty }}{e^{-s{\xi ^{2}}}}\big(1-\cos (h\xi )\big)\hspace{0.2778em}d\xi \end{array}\]
for every $h\in \mathbb{R}$. Applying Fubini’s Theorem and using the fact that the functions cosine and $\xi \longmapsto \frac{1-\cos (h\xi )}{{\xi ^{2}}}$ are even we get:
(38)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\int _{0}^{t}}ds{\int _{\mathbb{R}}}{\big[{p_{s}}(v+h)-{p_{s}}(v)\big]^{2}}dv& \displaystyle =& \displaystyle \frac{1}{\pi }{\int _{-\infty }^{\infty }}\Bigg[{\int _{0}^{t}}{e^{-s{\xi ^{2}}}}\hspace{0.2778em}ds\Bigg]\hspace{0.2778em}\big(1-\cos (h\xi )\big)\hspace{0.2778em}d\xi \\ {} & \displaystyle =& \displaystyle \frac{1}{\pi }{\int _{-\infty }^{\infty }}\big(1-{e^{-t{\xi ^{2}}}}\big)\frac{1-\cos (h\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi \\ {} & \displaystyle =& \displaystyle \frac{2}{\pi }{\int _{0}^{\infty }}\big(1-{e^{-t{\xi ^{2}}}}\big)\frac{1-\cos (|h|\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi .\end{array}\]Suppose that $h\ne 0$. By a simple change of variables in (38), using the fact that
we obtain
The function $g:\xi \longmapsto \frac{1-\cos (\xi )}{{\xi ^{2}}}$ is continuous on the interval $(0,+\infty )$ and consequently it is locally integrable. In addition, on the one hand, ${\lim \nolimits_{\xi \to 0}}g(\xi )=\frac{1}{2}$, which implies that g is integrable in a neighbourhood of 0. On the other hand, $|g(\xi )|\le \frac{2}{{\xi ^{2}}}$ for every $\xi >1$ and ${\textstyle\int _{1}^{+\infty }}\frac{1}{{\xi ^{2}}}d\xi <\infty $, which entails the integrability of g on a neighbourhood of $+\infty $. From all this, one can deduce that the integral ${\textstyle\int _{0}^{\infty }}\frac{1-\cos (\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi $ is convergent and, consequently, by using (39),
for every real $h\ne 0$. Morover, Inequality (40) is obviously true for $h=0$. Thus, (40) is statisfied for every $h\in \mathbb{R}$.
(39)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\int _{0}^{t}}ds{\int _{\mathbb{R}}}{\big[{p_{s}}(v+h)-{p_{s}}(v)\big]^{2}}dv& \displaystyle =& \displaystyle \hspace{2.5pt}\frac{2\hspace{0.1667em}|h|}{\pi }{\int _{0}^{\infty }}\big(1-{e^{-t\frac{{\xi ^{2}}}{{h^{2}}}}}\big)\frac{1-\cos (\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi \\ {} & \displaystyle \leqslant & \displaystyle \frac{2\hspace{0.1667em}|h|}{\pi }{\int _{0}^{\infty }}\frac{1-\cos (\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi .\end{array}\](40)
\[ {\int _{0}^{t}}ds{\int _{\mathbb{R}}}{\big[{p_{s}}(v+h)-{p_{s}}(v)\big]^{2}}dv\leqslant C|h|,\]This and Inequality (35) imply that
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{+}}(u,y,z)-{E^{+}}(u,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du& \displaystyle \le & \displaystyle C\big|\big|f(y)\big|-\big|f(x)\big|\big|\\ {} & \displaystyle \le & \displaystyle C\big|f(y)-f(x)\big|\\ {} & \displaystyle \le & \displaystyle C|y-x|,\end{array}\]
where in the last inequality we used Lemma 3. □Lemma 5.
For every $A>0$ and $t\in [0,T]$, there exists a positive constant c such that
\[ {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\geqslant \hspace{0.1667em}c\hspace{0.2778em}|y-x|\]
and
\[ {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\geqslant \hspace{0.1667em}c\hspace{0.2778em}|y-x|\]
for every $x,y\in [0,A]$.
Proof.
We present the proof just for the first inequality. The second is obtained in the same way. We have
where in the last inequality we used the fact that
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(f(z)-f(y))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(f(z)-f(x))^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}{\int _{0}^{\infty }}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(z/\sqrt{{a_{2}}}-f(y))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(z/\sqrt{{a_{2}}}-f(x))^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & & \displaystyle +{\int _{0}^{t}}{\int _{-\infty }^{0}}\bigg[\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(z/\sqrt{{a_{1}}}-f(y))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{(z/\sqrt{{a_{1}}}-f(x))^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du.\end{array}\]
Applying the changes of variables $Z=z/\sqrt{{a_{2}}}-f(x)$ in the first integral and $Z=z/\sqrt{{a_{1}}}-f(x)$ in the second one we obtain
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}{\int _{\mathbb{R}}}{\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|^{2}}\hspace{0.1667em}dz\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle \sqrt{{a_{2}}}\hspace{0.2778em}{\int _{0}^{t}}{\int _{-f(x)}^{\infty }}\bigg[\frac{1}{\sqrt{2u}}\exp \bigg(-\frac{{(Z-(f(y)-f(x)))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{Z^{2}}}{2u}\bigg)\bigg]{^{2}}dZ\hspace{0.1667em}du\\ {} & \displaystyle +& \displaystyle \sqrt{{a_{1}}}\hspace{0.2778em}{\int _{0}^{t}}{\int _{\infty }^{-f(x)}}\bigg[\frac{1}{\sqrt{2u}}\exp \bigg(-\frac{{(Z-(f(y)-f(x)))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{Z^{2}}}{2u}\bigg)\bigg]{^{2}},dZ\hspace{0.1667em}du\\ {} & \displaystyle \geqslant & \displaystyle \min (\sqrt{{a_{1}}},\sqrt{{a_{2}}}){\int _{0}^{t}}{\int _{\mathbb{R}}}\bigg[\frac{1}{\sqrt{2u}}\exp \bigg(-\frac{{(Z-(f(y)-f(x)))^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\frac{1}{\sqrt{2\pi u}}\exp \bigg(-\frac{{Z^{2}}}{2u}\bigg)\bigg]{^{2}}\hspace{0.1667em}dZ\hspace{0.1667em}du\\ {} & \displaystyle =& \displaystyle \min (\sqrt{{a_{1}}},\sqrt{{a_{2}}}){\int _{0}^{t}}{\int _{\mathbb{R}}}{\big[{p_{u}}(Z-\tilde{K})-{p_{u}}(Z)\hspace{0.2778em}\big]^{2}}\hspace{0.1667em}dZ\hspace{0.2778em}du,\end{array}\]
where $\tilde{K}=f(y)-f(x)$ and ${p_{u}}$ is the heat kernel defined by (37). Without loss of generality we can suppose that $x<y$. So,
Applying the same technique as that used in (38), we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}{\big[{p_{u}}(Z-h)-{p_{u}}(Z)\hspace{0.2778em}\big]^{2}}\hspace{0.1667em}dZ\hspace{0.2778em}du\\ {} & \displaystyle =& \displaystyle \frac{1}{2\pi }{\int _{0}^{t}}ds{\int _{-\infty }^{\infty }}{\big|{e^{-s{\xi ^{2}}/2-i\xi h}}-{e^{-s{\xi ^{2}}/2}}\big|^{2}}\hspace{0.2778em}d\xi \\ {} & \displaystyle =& \displaystyle \frac{1}{\pi }{\int _{0}^{t}}ds{\int _{-\infty }^{\infty }}{e^{-s{\xi ^{2}}}}\big(1-\cos (h\xi )\big)\hspace{0.2778em}d\xi \\ {} & \displaystyle =& \displaystyle \frac{2}{\pi }{\int _{0}^{\infty }}\big(1-{e^{-t{\xi ^{2}}}}\big)\frac{1-\cos (|h|\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi \end{array}\]
for every $h\in \mathbb{R}$. Thus,
(41)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}ds{\int _{\mathbb{R}}}{\big[{p_{s}}(Z-\tilde{K})-{p_{s}}(Z)\big]^{2}}dy\\ {} & \displaystyle =& \displaystyle \frac{2}{\pi }{\int _{0}^{\infty }}\big(1-{e^{-t{z^{2}}}}\big)\frac{1-\cos (\tilde{K}z)}{{z^{2}}}\hspace{0.2778em}dz\\ {} & \displaystyle =& \displaystyle \frac{2}{\pi }{\int _{0}^{\frac{1}{\tilde{K}}}}\big(1-{e^{-t{z^{2}}}}\big)\frac{1-\cos (\tilde{K}z)}{{z^{2}}}\hspace{0.2778em}dz+\frac{2}{\pi }{\int _{\frac{1}{\tilde{K}}}^{\infty }}\big(1-{e^{-t{z^{2}}}}\big)\frac{1-\cos (\tilde{K}z)}{{z^{2}}}\hspace{0.2778em}dz\\ {} & \displaystyle \ge & \displaystyle \frac{2}{\pi }{\int _{\frac{1}{\tilde{K}}}^{\infty }}\big(1-{e^{-t{z^{2}}}}\big)\frac{1-\cos (\tilde{K}z)}{{z^{2}}}\hspace{0.2778em}dz,\end{array}\]
\[ \frac{2}{\pi }{\int _{0}^{\frac{1}{\tilde{K}}}}\big(1-{e^{-t{z^{2}}}}\big)\frac{1-\cos (\tilde{K}z)}{{z^{2}}}\hspace{0.2778em}dz\ge 0.\]
Since $1-{e^{-t{z^{2}}}}\ge 1-{e^{-t{\tilde{K}^{-2}}}}$ for every $z\ge \frac{1}{\tilde{K}}$, from (41) we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\int _{0}^{t}}ds{\int _{\mathbb{R}}}{\big[{p_{s}}(Z-\tilde{K})-{p_{s}}(Z)\big]^{2}}dy& \displaystyle \ge & \displaystyle \frac{2}{\pi }\big(1-{e^{-t{\tilde{K}^{-2}}}}\big){\int _{\frac{1}{\tilde{K}}}^{\infty }}\frac{1-\cos (\tilde{K}z)}{{z^{2}}}\hspace{0.2778em}dz\\ {} & \displaystyle \ge & \displaystyle \tilde{K}\frac{2}{\pi }\big(1-{e^{-t{\tilde{K}^{-2}}}}\big){\int _{1}^{\infty }}\frac{1-\cos (\xi )}{{\xi ^{2}}}\hspace{0.2778em}d\xi ,\end{array}\]
where the last inequality is obtained after applying the change of variables $\xi =\tilde{K}z$.4.2 Proof of Theorem 6
Since Equation (7) is a particular case of (5), by Theorems 4 and 5, to get Theorem 6, it suffices to show that the fundamental solution associated to the operator ${\mathcal{L}_{p}}$ satisfies Conditions ${H_{i}}([0,1])$, for $i=1,2,3$. Consider $x,y\in [0,1];\hspace{0.2778em}x<y$ and $t\in [0,T]$.
4.2.1 Proof of ${H_{1}}([0,1])$
We have
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\big\| {\Delta _{y-x}}G(t-s,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}\\ {} & \displaystyle =& \displaystyle \hspace{0.1667em}{\int _{0}^{t}}\frac{1}{2\pi (t-s)}\bigg\{{\int _{\mathbb{R}}}\frac{1}{A(z)}\big[\big({A^{-}}(t-s,y,z)-{A^{-}}(t-s,x,z)\big)\hspace{0.1667em}{\mathbf{1}_{\{z\leqslant 0\}}}\\ {} & & \displaystyle +\big({A^{+}}(t-s,y,z)-{A^{+}}(t-s,x,z)\big)\hspace{0.1667em}{\mathbf{1}_{\{z>0\}}}\big]{^{2}}\hspace{0.1667em}dz\bigg\}\hspace{0.1667em}ds\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}\frac{1}{2\pi (t-s)}\bigg\{{\int _{\mathbb{R}}}\frac{1}{A(z)}\big[\big({E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big)\\ {} & & \displaystyle +\hspace{0.1667em}\beta \hspace{0.1667em}sign(z)\hspace{0.1667em}\big({E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big)\big]{^{2}}\hspace{0.2778em}dz\bigg\}\hspace{0.1667em}ds\\ {} & \displaystyle \geqslant & \displaystyle \hspace{0.1667em}\min \bigg(\frac{1}{{a_{1}}},\frac{1}{{a_{2}}}\bigg)\hspace{0.1667em}{\int _{0}^{t}}\frac{1}{2\pi (t-s)}\bigg\{{\int _{\mathbb{R}}}\big|\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|\\ {} & & \displaystyle -|\beta |\hspace{0.1667em}\big|{E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big|\big|{^{2}}\hspace{0.2778em}dz\bigg\}\hspace{0.1667em}ds.\end{array}\]
According to [6, page 54], we know that
\[ \| f-g{\| _{{L^{2}}(\mathbb{R})}^{2}}\hspace{0.2778em}\geqslant \hspace{0.1667em}\frac{1}{4}\hspace{0.1667em}{\big(\| f{\| _{{L^{2}}(\mathbb{R})}}+\| g{\| _{{L^{2}}(\mathbb{R})}}\big)^{2}}\hspace{0.2778em}{\bigg\| \frac{f}{\| f{\| _{{L^{2}}(\mathbb{R})}}}-\frac{g}{\| g{\| _{{L^{2}}(\mathbb{R})}}}\bigg\| _{{L^{2}}(\mathbb{R})}^{2}}\]
for every $f,g\in {L^{2}}(\mathbb{R});f\ne 0$ and $g\ne 0$ a.e. Thus,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\big\| {\Delta _{y-x}}G(t-s,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}\\ {} & \displaystyle \geqslant & \displaystyle \frac{1}{4}\hspace{0.1667em}\min \bigg(\frac{1}{{a_{1}}},\frac{1}{{a_{2}}}\bigg)\hspace{0.1667em}{\int _{0}^{t}}\frac{I(s)}{2\pi (t-s)}\\ {} & & \displaystyle \times \big(\big\| {E^{-}}(t-s,y,.)-{E^{-}}(t-s,x,.)\big\| \\ {} & & \displaystyle \hspace{1em}+\hspace{0.1667em}|\beta |\hspace{0.1667em}\big\| {E^{+}}(t-s,y,.)-{E^{+}}(t-s,x,.)\big\| \big){^{2}}ds,\end{array}\]
where
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle I(s)& \displaystyle =& \displaystyle \bigg\| \frac{|{E^{-}}(t-s,y,.)-{E^{-}}(t-s,x,.)|}{\| {E^{-}}(t-s,y,.)-{E^{-}}(t-s,x,.)\| }\\ {} & & \displaystyle -|\beta |\hspace{0.1667em}\frac{|{E^{+}}(t-s,y,.)-{E^{+}}(t-s,x,.)|}{\| {E^{+}}(t-s,y,.)-{E^{+}}(t-s,x,.)\| }\bigg\| {^{2}},\end{array}\]
and $\| .\| $ denotes $\| .{\| _{{L^{2}}(\mathbb{R})}}$. On the one hand we have
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}I(s)=1+{\beta ^{2}}\\ {} & & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}-\frac{2\hspace{0.1667em}|\beta |}{\| {E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\| \hspace{0.1667em}\| {E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\| }\\ {} & & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\times {\int _{\mathbb{R}}}\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|\hspace{0.2778em}\big|{E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big|\hspace{0.1667em}dz.\end{array}\]
On the other hand, applying Hölder’s Inequality, we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle \hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}\hspace{-0.1667em}{\int _{\mathbb{R}}}\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|\hspace{0.2778em}\hspace{0.1667em}\big|{E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big|\hspace{0.2778em}dz\\ {} & & \displaystyle \hspace{1em}\le \big\| {E^{-}}(t-s,y,.)-{E^{-}}(t-s,x,.)\big\| \hspace{0.1667em}\big\| {E^{+}}(t-s,y,.)-{E^{+}}(t-s,x,.)\big\| .\end{array}\]
Hence,
and therefore,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\big\| {\Delta _{y-x}}G(t-s,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}\\ {} & \displaystyle \geqslant & \displaystyle \hspace{0.1667em}\frac{{(1-|\beta |)^{2}}}{4}\hspace{0.1667em}\min \bigg(\frac{1}{{a_{1}}},\frac{1}{{a_{2}}}\bigg)\hspace{0.1667em}{\int _{0}^{t}}\frac{1}{2\pi (t-s)}\\ {} & & \displaystyle \times \big(\big\| {E^{-}}(t-s,y,.)-{E^{-}}(t-s,x,.)\big\| \\ {} & & \displaystyle +\hspace{0.1667em}|\beta |\hspace{0.1667em}\big\| {E^{+}}(t-s,y,.)-{E^{+}}(t-s,x,.)\big\| \big){^{2}}ds\\ {} & \displaystyle \geqslant & \displaystyle \frac{{(1-|\beta |)^{2}}}{4}\hspace{0.1667em}\min \bigg(\frac{1}{{a_{1}}},\frac{1}{{a_{2}}}\bigg)\hspace{0.1667em}{\int _{0}^{t}}\frac{1}{2\pi u}\\ {} & & \displaystyle \times \hspace{0.2778em}\big({\big\| {E^{-}}(u,y,.)-{E^{-}}(u,x,.)\big\| ^{2}}+\hspace{0.1667em}{\beta ^{2}}\hspace{0.1667em}{\big\| {E^{+}}(u,y,.)-{E^{+}}(u,x,.)\big\| ^{2}}\big)du,\end{array}\]
where in the last inequality we used the fact that ${x^{2}}+{y^{2}}\le {(x+y)^{2}}$ for every non-negative real numbers x and y.This and Lemma 5 show that Hypothesis ${H_{1}}([0,1])$ is satisfied.
4.2.2 Proof of ${H_{2}}([0,1])$
Using the expressions of ${A^{-}}$ and ${A^{+}}$ given in (30) we get
where
(42)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\big\| {\Delta _{y-x}}G(t-.,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}{\big|G(t-s,y,z)-G(t-s,x,z)\big|^{2}}\hspace{0.1667em}ds\hspace{0.1667em}dz\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}\Bigg[\frac{1}{2{a_{1}}\pi (t-s)}{\int _{-\infty }^{0}}{\big|{A^{-}}(t-s,y,z)-{A^{-}}(t-s,x,z)\big|^{2}}dz\Bigg]ds\\ {} & & \displaystyle +{\int _{0}^{t}}\Bigg[\frac{1}{2{a_{2}}\pi (t-s)}{\int _{0}^{\infty }}{\big|{A^{+}}(t-s,y,z)-{A^{+}}(t-s,x,z)\big|^{2}}dz\Bigg]\hspace{0.1667em}ds\\ {} & \displaystyle \le & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi (t-s)}{\int _{-\infty }^{0}}{\Delta _{max}}(s,z)dzds\\ {} & & \displaystyle +{\int _{0}^{t}}\frac{1}{2\pi (t-s)}{\int _{0}^{\infty }}{\Delta _{max}}(s,z)dz\hspace{0.1667em}ds,\end{array}\]
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\Delta _{\max }}(s,z)& \displaystyle =& \displaystyle \max \bigg(\frac{1}{{a_{1}}}\hspace{0.2778em}{\big|{A^{-}}(t-s,y,z)-{A^{-}}(t-s,x,z)\big|^{2}},\\ {} & & \displaystyle \frac{1}{{a_{2}}}\hspace{0.2778em}{\big|{A^{+}}(t-s,y,z)-{A^{+}}(t-s,x,z)\big|^{2}}\bigg)\\ {} & \displaystyle =& \displaystyle \max \bigg(\frac{1}{{a_{1}}}\big(\big({E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big)\\ {} & & \displaystyle -\beta \big({E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big)\big){^{2}},\\ {} & & \displaystyle \frac{1}{{a_{2}}}\big(\big({E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big)\\ {} & & \displaystyle +\beta \big({E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big)\big){^{2}}\bigg)\end{array}\]
for every $t\in (0,T]$ and $(x,y)\in {[0,1]^{2}};\hspace{0.2778em}y>x$.Since
\[ \max \big({\gamma _{1}}{(a-b)^{2}},{\gamma _{2}}{(a+b)^{2}}\big)\leqslant 2\max ({\gamma _{1}},{\gamma _{2}})\big({a^{2}}+{b^{2}}\big)\]
for any $(a,b)\in {\mathbb{R}^{2}}$ and any ${\gamma _{1}},{\gamma _{2}}>0$, we have
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {\Delta _{max}}(s,z)& \displaystyle \le & \displaystyle 2\max \bigg(\frac{1}{{a_{1}}},\frac{1}{{a_{2}}}\bigg)\big({\big|{E^{-}}(t-s,y,z)-{E^{-}}(t-s,x,z)\big|^{2}}\\ {} & & \displaystyle +{\beta ^{2}}{\big|{E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big|^{2}}\big).\end{array}\]
Thus,
(43)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\big\| {\Delta _{y-x}}G(t-.,x,.)\big\| _{{L^{2}}([0,t]\times \mathbb{R})}^{2}}\\ {} & \displaystyle \le & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi (t-s)}{\int _{\mathbb{R}}}{\Delta _{max}}(s,z)dzds\\ {} & \displaystyle \le & \displaystyle 2\max \bigg(\frac{1}{{a_{1}}},\frac{1}{{a_{2}}}\bigg)\Bigg[{\int _{0}^{t}}\bigg[\frac{1}{2\pi (t-s)}{\int _{\mathbb{R}}}\big|{E^{-}}(t-s,y,z)\\ {} & & \displaystyle -{E^{-}}(t-s,x,z)\big|{^{2}}dz\bigg]ds\\ {} & & \displaystyle +{\beta ^{2}}\hspace{0.1667em}{\int _{0}^{t}}\bigg[\frac{1}{2\pi (t-s)}{\int _{\mathbb{R}}}{\big|{E^{+}}(t-s,y,z)-{E^{+}}(t-s,x,z)\big|^{2}}dz\bigg]ds\Bigg].\hspace{21.0pt}\end{array}\]This and Lemma 4 show that Condition ${H_{2}}([0,1])$ is also satisfied.
4.2.3 Proof of ${H_{3}}([0,1])$
Consider $x,{x^{\prime }}\in [0,1]$ and $h>0$. We have
Using the expression of ${A^{-}}$ (30), denoting $\tilde{x}=\frac{x}{\sqrt{{a_{2}}}}$, ${\tilde{x}^{\prime }}=\frac{{x^{\prime }}}{\sqrt{{a_{2}}}}$ and $\tilde{h}=\frac{h}{\sqrt{{a_{2}}}}$, then making the change of variables ${z^{\prime }}=\frac{z}{\sqrt{{a_{1}}}}$, we get
(44)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}{\Delta _{h}}G(t-s,x,z)\hspace{0.1667em}{\Delta _{h}}G\big(t-s,{x^{\prime }},z\big)\hspace{0.1667em}dz\hspace{0.1667em}ds\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}{\int _{\mathbb{R}}}\big(G(t-s,x+h,z)-G(t-s,x,z)\big)\\ {} & & \displaystyle \big(G\big(t-s,{x^{\prime }}+h,z\big)-G\big(t-s,{x^{\prime }},z\big)\big)dz\hspace{0.1667em}ds\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}\Bigg[\frac{{m^{2}}(t-s)}{{a_{1}}}{\int _{-\infty }^{0}}\big({A^{-}}(t-s,x+h,z)-{A^{-}}(t-s,x,z)\big)\\ {} & & \displaystyle \big({A^{-}}\big(t-s,{x^{\prime }}+h,z\big)-{A^{-}}\big(t-s,{x^{\prime }},z\big)\big)dz\Bigg]ds\\ {} & \displaystyle +& \displaystyle {\int _{0}^{t}}\Bigg[\frac{{m^{2}}(t-s)}{{a_{2}}}{\int _{0}^{\infty }}\big({A^{+}}(t-s,x+h,z)-{A^{+}}(t-s,x,z)\big)\\ {} & & \displaystyle \big({A^{+}}\big(t-s,{x^{\prime }}+h,z\big)-{A^{+}}\big(t-s,{x^{\prime }},z\big)\big)dz\Bigg]ds\\ {} & \displaystyle =& \displaystyle L+K.\end{array}\]
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle L& \displaystyle =& \displaystyle {\int _{0}^{t}}\Bigg[\frac{{(1-\beta )^{2}}}{2\pi {a_{1}}u}{\int _{-\infty }^{0}}\big({E^{-}}(u,x+h,z)-{E^{-}}(u,x,z)\big)\\ {} & & \displaystyle \big({E^{-}}\big(u,{x^{\prime }}+h,z\big)-{E^{-}}\big(u,{x^{\prime }},z\big)\big)dz\Bigg]du\\ {} & \displaystyle =& \displaystyle \frac{1}{\sqrt{{a_{2}}}}{\int _{0}^{t}}\sqrt{\frac{{a_{2}}}{{a_{1}}}}\frac{{(1-\beta )^{2}}}{\hspace{0.1667em}2\pi u}\Bigg[{\int _{-\infty }^{0}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)\\ {} & & \displaystyle -\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}\hspace{0.1667em}\Bigg]du.\end{array}\]
Now, using the expression of ${A^{+}}$ (30) and making the change of variable ${z^{\prime }}=\frac{z}{\sqrt{{a_{2}}}}$, the integral K can be written in the form
where
\[ K=\hspace{0.1667em}{\int _{0}^{t}}\frac{1}{2\sqrt{{a_{2}}}\hspace{0.1667em}\pi u}\big[{K_{1}}+\beta \hspace{0.1667em}{K_{2}}+\beta \hspace{0.1667em}{K_{3}}+{\beta ^{2}}\hspace{0.1667em}{K_{4}}\big]\hspace{0.2778em}du,\]
where
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {K_{1}}& \displaystyle =& \displaystyle {\int _{0}^{+\infty }}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }},\\ {} \displaystyle {K_{2}}& \displaystyle =& \displaystyle {\int _{0}^{+\infty }}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}+{\tilde{x}^{\prime }}+\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}+{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }},\\ {} \displaystyle {K_{3}}& \displaystyle =& \displaystyle {\int _{0}^{+\infty }}\bigg(\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x}+\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x})^{2}}}{2u}\bigg)\bigg)\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }},\\ {} \displaystyle {K_{4}}& \displaystyle =& \displaystyle {\int _{0}^{+\infty }}\bigg(\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x}+\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x})^{2}}}{2u}\bigg)\bigg)\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}+{\tilde{x}^{\prime }}+\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}+{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}.\end{array}\]
By using the change of variable $z=-{z^{\prime }}$, we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {K_{2}}+{K_{3}}& \displaystyle =& \displaystyle {\int _{\mathbb{R}}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x}+\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x})^{2}}}{2u}\bigg)\bigg)\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}\end{array}\]
and
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {K_{1}}+{\beta ^{2}}\hspace{0.1667em}{K_{4}}\\ {} & \displaystyle =& \displaystyle {\int _{\mathbb{R}}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}\\ {} & & \displaystyle +\big({\beta ^{2}}-1\big){\int _{-\infty }^{0}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\hspace{0.1667em}\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}.\end{array}\]
Therefore,
(45)
\[ L+K={L_{1}}+\beta {L_{2}}+\bigg(\sqrt{\frac{{a_{2}}}{{a_{1}}}}{(1-\beta )^{2}}+{\beta ^{2}}-1\bigg){L_{3}},\]
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {L_{1}}& \displaystyle =& \displaystyle \hspace{0.1667em}{\int _{0}^{t}}\frac{du}{2\sqrt{{a_{2}}}\hspace{0.1667em}\pi u}\hspace{0.1667em}{\int _{\mathbb{R}}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\hspace{0.1667em}\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }},\\ {} \displaystyle {L_{2}}& \displaystyle =& \displaystyle {\int _{0}^{t}}\frac{du}{2\sqrt{{a_{2}}}\hspace{0.1667em}\pi u}{\int _{\mathbb{R}}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x}+\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}+\tilde{x})^{2}}}{2u}\bigg)\bigg)\hspace{0.1667em}\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}\end{array}\]
and
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {L_{3}}& \displaystyle =& \displaystyle {\int _{0}^{t}}\frac{du}{2\sqrt{{a_{2}}}\hspace{0.1667em}\pi u}{\int _{-\infty }^{0}}\bigg(\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-\tilde{x})^{2}}}{2u}\bigg)\bigg)\hspace{0.1667em}\\ {} & & \displaystyle \times \bigg(\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }}-\tilde{h})^{2}}}{2u}\bigg)-\exp \bigg(-\frac{{({z^{\prime }}-{\tilde{x}^{\prime }})^{2}}}{2u}\bigg)\bigg)d{z^{\prime }}.\end{array}\]
We first investigate the sign of the third integral. On the one hand, ${z^{\prime }}\le 0$, $\tilde{x}\ge 0$ and $\tilde{h}\ge 0$; thus, by virtue of the fact that the function $x\longmapsto \exp (-{x^{2}})$ is increasing on the interval $(-\infty ,0]$, we see that ${L_{3}}\ge 0$. On the other hand, using the expression of β, given in (32), by the fact that ${\rho _{2}}\ge {\rho _{1}}$ (see Condition (33)), we get
Therefore,
Now we will calculate explicitely the integral ${L_{1}}$. With the notation
for every $x,y\in \mathbb{R}$ and $u>0$, ${L_{1}}$ can be written in the form
(47)
\[ \mathcal{T}(x,y,u):={\int _{\mathbb{R}}}\exp \bigg(-\frac{{(v-y)^{2}}}{2u}\bigg)\hspace{0.1667em}\exp \bigg(-\frac{{(v-x)^{2}}}{2u}\bigg)\hspace{0.1667em}dv,\hspace{28.45274pt}\]
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {L_{1}}& \displaystyle =& \displaystyle {\int _{0}^{t}}\frac{1}{2\sqrt{{a_{2}}}\pi u}\big\{\mathcal{T}\big(\tilde{x}+\tilde{h},{\tilde{x}^{\prime }}+\tilde{h},u\big)-\mathcal{T}\big(\tilde{x},{\tilde{x}^{\prime }}+\tilde{h},u\big)\\ {} & & \displaystyle -\mathcal{T}\big(\tilde{x}+\tilde{h},{\tilde{x}^{\prime }},u\big)+\mathcal{T}\big(\tilde{x},{\tilde{x}^{\prime }},u\big)\big\}\hspace{0.1667em}du.\end{array}\]
By the changes of variables $V=v-x$ and $W=\frac{y-x-2v}{2\sqrt{u}}$, we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \mathcal{T}(x,y,u)& \displaystyle =& \displaystyle {\int _{\mathbb{R}}}\exp \bigg(-\frac{{v^{2}}}{2u}\bigg)\exp \bigg(\frac{-{((y-x)-v)^{2}}}{2u}\bigg)\hspace{0.2778em}dv\\ {} & \displaystyle =& \displaystyle \exp \bigg(-\frac{{(y-x)^{2}}}{4u}\bigg)\hspace{0.1667em}{\int _{\mathbb{R}}}\exp \bigg(-\frac{{((y-x)-2v)^{2}}}{4u}\bigg)\hspace{0.1667em}dv\\ {} & \displaystyle =& \displaystyle \sqrt{\pi u}\hspace{0.1667em}\exp \bigg(-\frac{{(y-x)^{2}}}{4u}\bigg).\end{array}\]
Thus, applying an integration by parts then the change of variables $w=\frac{y-x}{2\sqrt{u}}$, we get
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}& & \displaystyle {\int _{0}^{t}}\frac{1}{2\pi u}\mathcal{T}(x,y,u)\hspace{0.1667em}du:={\int _{0}^{t}}\frac{1}{2\sqrt{\pi u}}\hspace{0.1667em}\exp \bigg(-\frac{{(y-x)^{2}}}{4u}\bigg)\hspace{0.1667em}du\\ {} & \displaystyle =& \displaystyle \sqrt{\frac{t}{\pi }}\hspace{0.1667em}\exp \bigg(-\frac{{(y-x)^{2}}}{4t}\bigg)-\frac{{(y-x)^{2}}}{4\sqrt{\pi }}{\int _{0}^{t}}{u^{-3/2}}\hspace{0.1667em}\exp \bigg(-\frac{{(y-x)^{2}}}{4u}\bigg)\hspace{0.1667em}du\\ {} & \displaystyle =& \displaystyle \sqrt{\frac{t}{\pi }}\hspace{0.1667em}\exp \bigg(-\frac{{(y-x)^{2}}}{4t}\bigg)-\hspace{0.1667em}\frac{1}{2}(y-x)\hspace{0.1667em}\mathbf{erfc}\bigg(\frac{y-x}{2\sqrt{t}}\bigg).\end{array}\]
Hence,
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle {L_{1}}& \displaystyle =& \displaystyle \sqrt{\frac{t}{{a_{2}}\pi }}\bigg\{2\hspace{0.1667em}\exp \bigg(-\frac{{({\tilde{x}^{\prime }}-\tilde{x})^{2}}}{4t}\bigg)-\exp \bigg(-\frac{{({\tilde{x}^{\prime }}-\tilde{x}+\tilde{h})^{2}}}{4t}\bigg)\\ {} & & \displaystyle -\exp \bigg(-\frac{{({\tilde{x}^{\prime }}-\tilde{x}-\tilde{h})^{2}}}{4t}\bigg)\bigg\}\\ {} & & \displaystyle -\frac{1}{2\sqrt{{a_{2}}}}\bigg\{2\hspace{0.1667em}\big({\tilde{x}^{\prime }}-\tilde{x}\big)\hspace{0.1667em}\mathbf{erfc}\bigg(\frac{{\tilde{x}^{\prime }}-\tilde{x}}{2\sqrt{t}}\bigg)-\big({\tilde{x}^{\prime }}-\tilde{x}+\tilde{h}\big)\hspace{0.1667em}\mathbf{erfc}\bigg(\frac{{\tilde{x}^{\prime }}-\tilde{x}+\tilde{h}}{2\sqrt{t}}\bigg)\hspace{0.1667em}\\ {} & & \displaystyle -\big({\tilde{x}^{\prime }}-\tilde{x}-\tilde{h}\big)\mathbf{erfc}\bigg(\frac{{\tilde{x}^{\prime }}-\tilde{x}-\tilde{h}}{2\sqrt{t}}\bigg)\bigg\}.\end{array}\]
The function $\tilde{h}\longmapsto {L_{1}}(\tilde{h})={L_{1}}$ is clearly twice differentiable and via a simple calculation we get
\[ {L^{\prime }_{1}}(\tilde{h})=-\frac{1}{2\sqrt{{a_{2}}}}\bigg\{\mathbf{erfc}\bigg(\frac{{\tilde{x}^{\prime }}-\tilde{x}-\tilde{h}}{2\sqrt{t}}\bigg)-\mathbf{erfc}\bigg(\frac{{\tilde{x}^{\prime }}-\tilde{x}+\tilde{h}}{2\sqrt{t}}\bigg)\bigg\}\]
and
\[ {L^{\prime\prime }_{1}}(\tilde{h})=\frac{-1}{2\sqrt{{a_{2}}}\sqrt{\pi t}}\bigg[\exp \bigg(-\frac{{({\tilde{x}^{\prime }}-\tilde{x}-h)^{2}}}{4t}\bigg)+\exp \bigg(-\frac{{({\tilde{x}^{\prime }}-\tilde{x}+h)^{2}}}{4t}\bigg)\bigg].\]
It’s easy to check that ${L_{1}}(0)={L^{\prime }_{1}}(0)=0$ and that ${L^{\prime\prime }_{1}}$ is bounded. From all this and by Taylor’s formula, we obtain
for every $h>0$, where C denotes a positive universal constant.Applying the same techniques used in the above argunents, and since $\beta \ge 0$ (see the expression of β given in (32) and Assumption (33)), we get
for every $h>0$. Combining (44), (31), (46), (48) and (49), the proof of ${H_{3}}([0,1])$ is finished, and consequently, the proof of Theorem 6 is also finshed.
Remark 4.
Considering an integer $d\ge 1$, one can extend Equation (5) to the d-dimensional case by introducing the following SPDE:
with
where $x\longmapsto {R_{ij}}(x)$ and $x\longmapsto {r_{ij}}(x)$ are two measurable and bounded real-valued functions satisfying
and there exists a constant $\nu >0$ such that
for every $x\in {\mathbb{R}^{d}}$, $\xi =({\xi _{1}},\dots ,{\xi _{d}})\in {\mathbb{R}^{d}}$ and $i,j\in \{1,\dots ,d\}$. In (52), $\| .{\| _{d}}$ denotes the Euclidean norm in ${\mathbb{R}^{d}}$, and in (51) $\frac{\partial }{\partial {x_{i}}}$ denotes the partial derivative in the distributional sense. The noise ${W_{d}}$ is a centered Gaussian field ${W_{d}}=\{{W_{d}}(t,C);t\in [0,T],C\in {B_{b}}({\mathbb{R}^{d}})\}$ with covariance
where ${\lambda _{d}}$ denotes the Lebesgue measure on ${\mathbb{R}^{d}}$ and ${B_{b}}({\mathbb{R}^{d}})$ is the set of ${\lambda _{d}}$-bounded Borel sub-sets of ${\mathbb{R}^{d}}$. In the particular case where $d=1$, SPDE (50) is clearly reduced to Equation (5).
(50)
\[ \left\{\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle \frac{\partial u(t,x)}{\partial t}& =& {\mathcal{L}_{d}}u(t,x)+{\dot{W}_{d}}(t,x);\hspace{1em}t>0,\hspace{0.2778em}x=({x_{1}},\dots ,{x_{d}})\in {\mathbb{R}^{d}},\\ {} u(0,.)& :=& 0,\hspace{0.2778em}\end{array}\right.\](51)
\[ {\mathcal{L}_{d}}={\sum \limits_{i,j=1}^{d}}\frac{1}{{r_{ij}}(x)}\frac{\partial }{\partial {x_{i}}}\bigg({R_{ij}}(x)\frac{\partial }{\partial {x_{j}}}\bigg),\](52)
\[ {r_{ij}}(x){\xi _{i}}{\xi _{j}}\ge \nu \| \xi {\| _{d}^{2}}\hspace{1em}\text{and}\hspace{1em}{R_{ij}}(x){\xi _{i}}{\xi _{j}}\ge \nu \| \xi {\| _{d}^{2}}\]According to [1], if we denote by ${G_{d}}$ the fundamental solution associated to the operator ${\mathcal{L}_{d}}$, then there exist two constants ${D_{1}}>0$ and ${D_{2}}>0$ such that
Denoting by I the right-hand side of Inequality (54), we have
where the second equality in (55) is obtained by the change of variables ${y^{\prime }_{i}}=({x_{i}}-{y_{i}})\sqrt{\frac{2{D_{2}}}{t-s}}$. Since the term ${\textstyle\int _{0}^{t}}\frac{ds}{{(t-s)^{d/2}}}$ is finite if, and only if $d<2$, from (54) and (55) we deduce that, for every $d\ge 2$ we have
\[ {G_{d}}(t,x,y)\ge \frac{{D_{1}}}{{t^{d/2}}}\exp \bigg(-\frac{{D_{2}}\| x-y{\| _{d}^{2}}}{t}\bigg)\]
for every $t\in [0,T]$ and $(x,y)\in {\mathbb{R}^{d}}$. It follows then that
(54)
\[ {\int _{0}^{t}}{\int _{{\mathbb{R}^{d}}}}{G_{d}^{2}}(t-s,x,y)dyds\ge {\int _{0}^{t}}{\int _{{\mathbb{R}^{d}}}}\frac{{D_{1}^{2}}}{{(t-s)^{d}}}\exp \bigg(-\frac{2{D_{2}}\| x-y{\| _{d}^{2}}}{t-s}\bigg)dyds.\](55)
\[\begin{array}{r@{\hskip10.0pt}c@{\hskip10.0pt}l}\displaystyle I& \displaystyle =& \displaystyle {\int _{0}^{t}}\frac{{D_{1}^{2}}}{{(t-s)^{d}}}{\prod \limits_{i=1}^{d}}\bigg({\int _{\mathbb{R}}}\exp \bigg(-\frac{2{D_{2}}{({x_{i}}-{y_{i}})^{2}}}{t-s}\bigg)d{y_{i}}\bigg)ds\\ {} & \displaystyle =& \displaystyle {\int _{0}^{t}}\frac{{D_{1}^{2}}}{{(t-s)^{d}}}{\bigg(\sqrt{\frac{\pi (t-s)}{2{D_{2}}}}\bigg)^{d}}ds\\ {} & \displaystyle =& \displaystyle {D_{1}^{2}}{\bigg(\frac{\pi }{2{D_{2}}}\bigg)^{d/2}}{\int _{0}^{t}}\frac{ds}{{(t-s)^{d/2}}},\end{array}\]Therefore, for $d\ge 2$, the Wiener integral ${\textstyle\int _{0}^{t}}{\textstyle\int _{{\mathbb{R}^{d}}}}{G_{d}}(t-s,x,y){W_{d}}(ds,dy)$ is not well-defined and consequently, the mild solution to Equation (50) exists if, and only if $d=1$.
