Abstract
In this paper, we consider the stochastic heat equation of the form
where W is a fractional Brownian sheet, \(\Delta+\Delta_{\alpha}\) is a pseudo differential operator on \({\mathbb {R}}\) which gives rise to a Lévy process consisting of the sum of a Brownian motion and an independent symmetric α-stable process, and \(f:[0,T]\times\mathbb{R}\times\mathbb {R}\rightarrow\mathbb{R}\) is a nonlinear measurable function. We introduce the existence, uniqueness, Hölder regularity and density estimate of the solution.
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1 Introduction
Stochastic heat equations and fractional heat equations driven by fractional Brownian motion (sheet) are a recent research direction in probability theory and its applications. In Balan and Conus [1], Song [2], the authors considered intermittency for the fractional heat equation and a class of stochastic partial differential equations. In Chen et al. [3], Hu et al. [4], Hu, Lu and Nualart [5],the authors discussed the Feynman-Kac formula for fractional heat equations. In Bo et al. [6], Diop and Huang [7], Duncan et al. [8], Balan [9], Hu and Nualart [10], Liu and Yan [11], the authors introduced the stochastic heat equations with fractional white noises, and about the stochastic heat equations with fractional-colored noises we can see Jiang et al. [12, 13], Balan and Tudor [14, 15], Tudor [16] and the references therein. However, it is very limited to study the stochastic heat equations driven by the mixed fractional operator \(\Delta+\Delta_{\alpha}\) and fractional Brownian sheet, where \(\Delta_{\alpha}=-(-\Delta)^{\alpha/2}\) is the fractional power of the Laplacian. On the other hand, many mathematical problems in physics and engineering with respect to systems and processes are represented by a kind of equations, more precisely fractional order differential equations driven by fractional noise. The increasing interest in this class of equations is motivated both by their applications to fluid dynamic traffic model, viscoelasticity, heat conduction in materials with memory, electrodynamics with memory and also because they can be employed to approach nonlinear conservation laws (see, for example, Sobczyk [17] and Droniou and Imbert [18]). Therefore, it seems interesting to handle the mixed fractional heat equations driven by fractional Brownian sheet. In this paper, we are concerned with the stochastic heat equation of the form
with \(0<\alpha<2\), where \({W}(t,x)\) is the fractional Brownian sheet and the nonlinear measurable function \(f:[0,T]\times \mathbb{R}\times\mathbb{R}\to\mathbb{R}\) and the initial-value \({\vartheta}(x)\) satisfy the following assumptions:
Assumption 1
For some \(p\geq2\), we have
and there is a constant \(\theta \in(0, 1)\) with \(p\theta < 1\) such that
Assumption 2
For each \(T>0\), there exists a constant \(C>0\) such that
for all \((t,x,y)\in[0,T]\times\mathbb{R}\times\mathbb{R}\) and \(x',y'\in\mathbb{R}\).
The paper is organized as follows. Section 2 contains some preliminaries on the pseudo differential operator \(\Delta+\Delta_{\alpha}\), the double-parameter fractional noises and the related Malliavin calculus. In Section 3, we study the existence and uniqueness of the mild solution to (1.1) by using a Picard approximation. In Section 4 we show the Hölder regularity of the solution \(u(t, x)\). Section 5 is devoted to showing the existence of the density of \(u(t, x)\) and we show that the law of \(u(t, x)\) is absolutely continuous with respect to the Lebesgue measure on \({\mathbb {R}}\) by using Malliavin calculus.
2 Preliminaries
In this section, we briefly recall some basic results for Green function of the pseudo differential operator \(\Delta+\Delta_{\alpha}\) and Malliavin calculus associated with fractional Brownian sheet. We refer to Chen et al. [19–23] and Nualart [24] and the references therein for more details. For convenience, in this paper we assume that C is a positive constant depending only on T, H, α and its value may be different in different positions.
2.1 On the pseudo differential operator \(\Delta +\Delta _{\alpha}\)
It is well known that, for a second order elliptic differential operator \({\mathscr {D}}\) on \({\mathbb {R}}^{d}\) satisfying some natural conditions, there is a diffusion process X on \({\mathbb {R}}^{d}\) such that \({\mathscr {D}}\) is its infinitesimal generator, and its transition density function is the fundamental solution of the equation
We also call the fundamental solution the heat kernel (Green function) of \({\mathscr {D}}\). For a large class of Markov processes with discontinuous sample paths, we also have such a correspondence, and such Markov processes have been widely used in various problems. In this one, an important Markov processes with discontinuous sample paths is (rotationally) symmetric α-stable (Lévy) process with \(0<\alpha\leq2\).
A symmetric α-stable process \(X=\{X_{t},t\geq0, {\mathbb {P}}_{x}, x\in{\mathbb {R}}\}\) on \({\mathbb {R}}^{d}\) is a Lévy process such that
for every \(x,\zeta\in\mathbb{R}^{d}\), where \(\mathbb{E}_{x}\) denotes the expectation with respect to \({\mathbb {P}}_{x}\). When \(\alpha=2\), X is a Brownian motion on \(\mathbb{R}^{d}\) whose infinitesimal generator is the Laplacian Δ. When \(0<\alpha<2\), the infinitesimal generator of a symmetric α-stable process X is the fractional Laplacian \(\Delta_{\alpha}=-(-\Delta)^{\alpha/2}\), which is a nonlocal operator and it can be defined by
where \(\mathcal{L}(d,\alpha):=\alpha2^{\alpha-1}\pi^{-\frac{d}{2}} \Gamma(\frac{d+\alpha}{2})/{\Gamma(1-\frac{\alpha}{2})}\) and Γ denotes the classical Gamma function. In this paper, we consider only the case \(d=1\).
Let now \(X^{\alpha}\) be a real value α-stable process with \(0<\alpha<2\) and let B be a real value Brownian motion independent of \(X^{\alpha}\). Define the process X by
Then the infinitesimal generator of X is \(\Delta +\Delta _{\alpha}\) and
for every \(x,\zeta\in\mathbb{R}^{d}\). Denote by \(G_{\alpha}(t,x)\) the fundamental solution of the equation
(or equivalently the heat kernel of \(\Delta +\Delta _{\alpha}\)). It follows from Chen et al. [23] that
for all \(t>s>0\), \(x,y\in{\mathbb {R}}^{d}\) and some constants \(C_{1},C_{2}>1\), where \(a_{1}\wedge a_{2}:=\min\{a_{1},a_{2}\}\) for \(a_{1},a_{2}\in{\mathbb {R}}\) and \(G_{\alpha}(s,y;t,x):=G_{\alpha}(t-s,x-y)\).
2.2 Malliavin calculus
Recall that a fractional Brownian sheet defined on a probability space \((\Omega,\mathscr{F}, P)\) with indices \(H_{1},H_{2}\in(0, 1)\), \(W=\{W(t, x), t\in[0,T], x\in\mathbb{R}\}\) is a Gaussian random field with \(W(0,0)=0\) and
for all \(s,t>0\), \(x,y\in{\mathbb {R}}\), where
Let \(\mathcal{H}\) be the completion of the linear space \({\mathcal {I}}\) generated by the indicator functions \(1_{(s,t]\times(x,y]}\) on \([0,T]\times\mathbb{R}\) with respect to the scalar product
The following embedding property follows from Bo et al. [6] (see also Jiang et al. [13] and Wei [25]).
Proposition 2.1
For \(H> \frac{1}{2}\) we have
Define the mapping \(W(g)\) between \({\mathcal {I}}\) and the Gaussian space associated with W by
Then it is an isometry and it can be extended to \({\mathcal {H}}\), which is called the Wiener integral of g with respect to W. Denote
for any \(0\leq s< t\leq T\) and \(x, y \in\mathbb{R}\).
Proposition 2.2
For \(\varphi, \psi\in\mathcal{H}\), we have \({E}[W(\mu)]=0\) and
Proposition 2.3
If \(H\in(\frac{1}{2},1)\) and \(\varphi, \psi\in L^{\frac{1}{H}}([a,b])\), then
Consider now the set \({\mathcal {C}}\) of smooth and cylindrical functional
where the function f and its derivatives of all orders are bounded and \(g_{i}\in\mathcal{H}\), \(i=1,\ldots, n\). Define the derivative operator DϜ (the Malliavin derivative) by
for the functional Ϝ of the form (2.2). Then D is a close operator from \(L^{2}(\Omega)\) into \(L^{2}(\Omega;{\mathcal {H}})\). Denote \(D_{h}\digamma=\langle D\digamma,h\rangle_{\mathcal{H}}\). Let \(\mathbb{D}_{h}\) and \(\mathbb{D}^{1,2}\) be the closures of \(\mathcal{C}\) with respect to the norms
for \(h\in\mathcal{H}\) and
respectively. Then \(\mathbb{D}^{1,2}\) is the domain of D on \(L^{2}(\Omega)\) and
for each \(n\in\mathbb{N}\), if \(\sum_{n=1}^{\infty}{E}|D_{h_{n}}\digamma|^{2}<+\infty\), where \(\{h_{n}, n\geq1\}\) is an orthogonal basis of \(\mathcal{H}\).
The divergence operator (integral) δ is defined as the adjoint of D. A random variable \(u\in L^{2}(\Omega;{\mathcal {H}})\) belongs to the domain \(\operatorname{Dom}(\delta)\) of δ, provided
for all \(\digamma\in\mathcal {C}\). Thus, \(\delta(u)\) can be determined by the next duality relationship:
We will also use the next notations:
and
By using Malliavin calculus for stochastic partial differential equations (abbr. SPDEs) driven by fractional noises, we can get the following propositions (see, e.g., Wei [25] and Jiang et al. [13]).
Proposition 2.4
Let \(\mathscr{F}_{N}:= \sigma\{W(M),M\subset N\}\) for \(N\in \mathscr{B}([0, T ]\times{\mathbb {R}})\), and let the random variable Y be square integrable. If Y is measurable with respect the σ-field \(\mathscr{F}_{N^{c}}\), then
almost surely.
Proposition 2.5
For a random variable Y belonging to \(\mathbb{D}^{1,2}\), if \(\|DY\|^{2}_{\mathcal{H}}>0\) almost surely, the law of Y is absolutely continuous with respect to the Lebesgue measure.
3 Existence and uniqueness of the solution
Given a filtered probability space \((\Omega,\mathscr{F}, (\mathscr{F}_{t})_{t\geq0}, P)\) with the natural filtration \((\mathscr {F}_{t})_{t\geq0}\) of W. In this section, the Cauchy problem (1.1) will be discussed. By using the heat kernel \(G_{\alpha}(s,y;t,x)\) of \(\Delta +\Delta _{\alpha}\), as usual (see, e.g., Walsh [26]) we say that the stochastic field
is a mild solution to (1.1) if
for all \(t\geq0\) and \(x\in{\mathbb {R}}\). Now we can state the main result in this section, and its proof could be derived by using some estimates of the heat kernel \({G_{\alpha}}(s,y;t,x)\) and some properties of the stochastic integral
Theorem 3.1
Under Assumptions 1 and 2, equation (3.1) admits a unique solution \(u=\{u(t,x),(t, x)\in[0,T]\times{\mathbb {R}}\}\) such that
for all \(\alpha\in(0,2)\) and \(p\geq2\).
Proof
We first use Picard’s approximation to get a solution to (3.1) and then we show that the solution is unique. This proof will be decomposed into three steps, and we define
for all \(t\geq0\), \(x\in{\mathbb {R}}\) and \(n\in \mathbb{N}=\{0,1,2,\ldots\}\).
Step I. We prove that
By Hölder’s inequality and Assumption 1, we get
for all \(p\geq2\). Notice that (2.1) implies that
We see that \(\sup_{t\in[0,T], x\in\mathbb {R}}{E}|u_{0}(t,x)|^{p}<+\infty\).
On the other hand, for each \(n\geq1\) and \(p\geq2\) we denote
By (3.2) it follows that
We need to estimate \(\Phi_{p,n}(t, x)\) and \(\Psi_{p,n}(t, x)\). Clearly, we have
and
for all \(t>s>0\). It follows that
which implies that
by Propositions 2.1, 2.2, and (2.1). Similarly, by the Hölder inequality we get
Denote \(g_{x}(t-s)=\int_{\mathbb{R}}\vert \frac{\partial{G_{\alpha}}}{\partial y}(s,y;t,x)\vert \, dy\), \({\mathbb {D}}_{x}= \{z\in{\mathbb {R}} \mid |x-z|< (t-s)^{\frac {3}{2(1+\alpha)}} \}\), and
for all \(t>s>0\) and \(x,y\in{\mathbb {R}}\). It follows from (3.6), (3.7), and (2.1) that
and
Combining this with Lemma 15 in Dalang [27], we get
Step II. We prove that \(\{u_{n}(t,x)\}_{n\in\mathbb{N}}\) converges in \(L^{p}(\Omega)\) for any \(p\geq2\). For \(n\geq2\), we have
and
Combining this with Gronwall’s inequality, we get
which implies that \(\{u_{n}(t,x)\}_{n\geq0}\) is a Cauchy sequence in \(L^{p}(\Omega)\). Define
in \(L^{p}(\Omega)\). Then we have
for each \((t,x)\in[0,T]\times{\mathbb {R}}\). Taking \(n\to+\infty\) in \(L^{p}(\Omega)\) for (3.2), we see that \(\{u(t,x): (t,x)\in [0,t]\times{\mathbb {R}}\}\) satisfies (3.1).
Step III. We prove the uniqueness of the solution. Let u and û be the two mild solutions of (1.1), then
It follows from Gronwall’s inequality that
for all \(T>0\). Thus, we have completed the proof of the theorem. □
4 Hölder regularity and p-variation of the solution
In this section we expound and prove the next theorem, which gives the Hölder regularity of the solution \(u=\{u(t,x),(t,x)\in[0, T]\times{\mathbb {R}}\}\) to (3.1).
Theorem 4.1
Let \(H_{1},H_{2} \in(\frac{1}{2},1)\) and \(\alpha\in(1,2)\). Under Assumptions 1 and 2, the solution \(u(t, x)\) has a continuous version which is γ-Hölder continuous in t with \(\gamma\in (0,\vartheta_{1} )\) and ν-Hölder continuous in x with \(\nu\in (0,\vartheta_{2} )\), where
In order to show that the theorem holds we need two lemmas.
Lemma 4.1
We have
for all \(0< r< t\leq T\), \(x\in\mathbb{R}\) and \(\theta_{1}\in(0,1)\). Moreover, when \(0<\theta_{1}<\frac{2H_{1}-1}{2}+\frac{H_{2}}{2}\), we have also
for all \(t\in[0,T]\) and \(x\in{\mathbb {R}}\).
Proof
Given \(t>r\) and \(z\in{\mathbb {R}}\). Recall that \({\mathbb {D}}_{x}= \{ y\in{\mathbb {R}} \mid |x-y|< (t-r)^{\frac{3}{2(1+\alpha)}} \}\),
for all \(0< r< t\leq T\) and \(x\in\mathbb{R}\). Clearly, we have
by the fact \(x^{2}e^{-x^{2}}\leq1\) and
for all \(t>r>0\) and \(x\in{\mathbb {R}}\). Thus, we have introduced (4.1) and hence (4.2) follows. □
Lemma 4.2
For all \(t>r\geq0\), \(0<\theta_{2}<H_{2}\), and \(x,z\in\mathbb{R}\), we have
and
Proof
Given \(t>r\) and \(z\in{\mathbb {R}}\). Recall that \({\mathbb {D}}_{z}= \{ x\in{\mathbb {R}} \mid |x-z|< (t-r)^{\frac{3}{2(1+\alpha)}} \}\). Then we have
Clearly, we have
by the fact \(|x|e^{-x^{2}}\leq C\) for all \(x\in{\mathbb {R}}\), and
for all \(0<\theta_{2}<H_{2}\). Thus, we have proved the estimate (4.3) and (4.4). □
Proof of Theorem 4.1
We shall divide the proof into two steps.
Step 1. We first consider the temporal case. Denote
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). Then we have
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). By Hölder’s inequality, the semigroup property and (3.2), we have
with \(p\theta<\alpha\). Some elementary calculations can show that
and
which gives
Let now us estimate the term \(A^{1}_{2}(t, s, x)\). Denote
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). We then have
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). Moreover, for every \(\theta_{1}\in(0,1)\) we let
with \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). Then we have
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). But, by using (3.2), Proposition 2.4, Lemma 4.1, and the mean-value theorem, we see that there is an ξ between s and t such that
for all \(0<{\theta_{1}}<\frac{2H_{1}-1}{2}+\frac{H_{2}}{2}\). Similarly, one can prove that
It follows that
for all \({\theta_{1}}\in(0,\vartheta_{1})\). On the other hand, we have
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\), which gives
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\) by (3.7). Combining this with (4.6), we get
for \({\theta_{1}} \in (0,\vartheta_{1} )\).
Finally, by the Hölder inequality, Assumption 2, Theorem 3.1, and (3.8), we have
and
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\). It follows that
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\).
Thus, we have obtained the desired estimate
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\), which implies that
for all \(x\in{\mathbb {R}}\) and \(0\leq s< t\leq T\) by taking \(\nu\in \min\{\theta,\theta_{1}\}\). This shows the Hölder continuity in time variables t by Gronwall’s inequality.
Step 2. We consider the spatial case. For all \(t\in[0, T]\) and \(x, y \in{\mathbb {R}}\), we need to estimate the following expressions:
We have
for all \(t\in[0,T]\) and \(x, y \in{\mathbb {R}}\) by Assumption 1. Denote
for all \(t\geq0\) and \(x,y\in{\mathbb {R}}\). Then we have
Similar to Step I, by using (3.2), Proposition 2.4, Lemma 4.2, and the mean-value theorem, one can see that
for all \({\theta_{2}}< H_{2}\). Similarly, one can also prove
for all \({\theta_{2}}< H_{2}\). It follows that
for all \({\theta_{2}}< H_{2}\). Finally, we consider the term \(|A^{2}_{3}(t,x,y)|\). By the Hölder inequality, Assumption 1, and (3.8), we have
Combining this with (4.13) and (4.14), we have
for all \(0<{\theta_{2}}<H_{2}\). Thus, we have proved the Hölder continuity in space variables x by Gronwall’s inequality. □
As an immediate result of the above theorem, we see that the quadratic variation is zero. At the end of this section, we give the p-variation of the solution. For convenience we consider the following special equation:
As in Section 3, the solution of (4.16) can be written in mild form as
where \(\bar{G}_{\alpha}(s,y;t,x)\) stands for the heat kernel of \(\Delta_{\alpha}\). It follows from Chen et al. [23] that
for all \(0 \leq s < t\), \(x, y\in{\mathbb {R}}\) and some constant \(C\geq0\).
Lemma 4.3
Let ū be the solution of (4.16). Then, for all \(0< s< t\), we have
where \(\eta \in(0, \frac{\alpha H_{1}+H_{2}-1}{\alpha})\).
Proof
Similar to the proof of Step 1 of Theorem 4.1 and Lemma 4.1, when \(\eta \in(0, \frac {\alpha H_{1}+H_{2}-1}{\alpha})\), we obtain
This completes the proof. □
For \(T>0\), let \(\tau_{n} =\{0 = t_{0} < t_{1} <\cdots< t_{n} = T\}\) be a partition of \([0, T]\) such that the mesh size \(|\Delta_{n}| =\max_{j} |t_{j}-t_{j-1}|\to0\) (\(n\to\infty\)). Recall that a process \(Y=\{Y_{t}; 0\leq t <\infty\}\) is of bounded p-variation with \(p\geq1\) on the interval \([0,T]\) if the limit of
exists in \(L^{1}(\Omega)\), as \(n\to\infty\), We denote by \(V^{p}(Y;T)\) the p-variation on \([0,T]\).
We new consider p-variations of the solution to the fractional heat equation (4.16).
Theorem 4.2
Let \((\bar{u}(t, x), t\in[0,T], x\in{\mathbb {R}})\) be given by (4.17). For \(H_{1}\in(\frac{1}{2},1)\), and \(H_{2}=\frac{1}{2}\), we have
if \(p>\frac{2\alpha}{2\alpha H_{1}-1}\) for all \(x\in{\mathbb {R}}\).
Proof
By Lemma 4.3, we have
which shows that \(t\mapsto\bar{u}(t,x)\) has p-variation 0 when \(p>\frac{2\alpha}{2\alpha H_{1}-1}\) for all \(x\in{\mathbb {R}}\). This completes the proof. □
5 Existence of the density
In this part, we will focus to prove the absolute continuity of the distribution of solution \(\{u(t, x):(t,x)\in[0,T]\times{\mathbb {R}}\}\) given in Section 3 by using Malliavin calculus.
Proposition 5.1
Under the assumptions in Theorem 3.1, if the function \((t,x,z)\mapsto f(t,x,z)\) and its partial derivatives of order 1 are bounded, then \(u(t, x)\in \mathbb{D}^{1,2}\) and
for all \(0\leq r\leq t\leq T\) and \(x,v\in{\mathbb {R}}\).
Proof
By approximating we can introduce the theorem. Let \(u_{n}(t,x)\) satisfy the next equation:
for all \(n=0,1,2,\ldots\) . Then \(u_{n}(t, x)\in\mathbb{D}_{h}\) and it satisfies
for each \(n\in\mathbb{N}\) and \(h\in\mathcal{H}\) (see the argument in Zhang and Zheng [28]). Since
in \(L^{p}\), there is a random field \(u^{(h)}(t, x)\) such that
uniformly on \((t, x)\in[0,T]\in{\mathbb {R}}\), and
It follows from the closeness of the operator \(D_{h}\) that \(D_{h}u(t, x)=u^{(h)}(t, x)\), \(u(t, x)\in\mathbb{D}_{h}\), and
for all \(0\leq t\leq T\) and \(x\in{\mathbb {R}}\). Now, we claim that \(u(t, x)\in\mathbb{D}^{1,2}\). By (5.2), we have
for all \(0\leq t\leq T\), \(x\in{\mathbb {R}}\) and \(\{h_{n}, n\geq1\} \subset{\mathcal{H}}\). Set
for all \(0\leq t\leq T\). Then we have
for all \(t\in[0,T]\) by (5.3) and Cauchy’s inequality, where we have used the fact that
for all \(s,t\in[0,T]\) and \(x\in{\mathbb {R}}\). It follows from Gronwall’s inequality that
Letting \(l\to+\infty\), we get
which shows that \(u(t,x)\in\mathbb{D}^{1,2}\) for all \(0\leq t\leq T\) and \(x\in{\mathbb {R}}\).
Finally, let us calculate the derivative \(Du(t,x)\) for all \((t, x)\in [0,T]\times{\mathbb {R}}\). Since \(u(t, x)\) is \(\mathscr{F}_{t}\)-adapted, by Proposition 2.4 there exists a measurable function \(D_{r,v}u(t, x)\in\mathcal{H}\) such that \(D_{r,v}u(t, x)= 0 \) if \(r> t\) and for any \(h\in\mathcal{H}\),
for all \(0\leq t\leq T\) and \(x\in{\mathbb {R}}\). From (5.2), (5.4), and the Fubini theorem, it follows that
for \((t, x)\in[0,T]\times{\mathbb {R}}\). Thus, we have proved the desired formula,
for all \(0\leq r\leq t\), \(x,v\in{\mathbb {R}}\), and the theorem follows. □
Theorem 5.2
Under the assumptions of Theorem 3.1 and assuming the function f and its partial derivatives of order 1 to be bounded, the distribution of the random variable \(u(t, x)\) is absolutely continuous with respect to the Lebesgue measure for all \((t,x)\in[0,T]\times {\mathbb {R}}\).
For proving Theorem 5.2, we will make use of the following lemma.
Lemma 5.1
Let \(t>0\) and \(0< r< t\). Denote
for \(s\in[t-r, t]\) and \(y\in{\mathbb {R}}\). Then we have
Proof
Let \(0< r< t\) and \(s\in[t-r, t]\). Then we have
by the proof of Proposition 5.1. Denote
for \(s\in[t-r, t]\) and \(y\in{\mathbb {R}}\). Then, by (5.1) and (3.2), we have
for \(s\in[t-r, t]\) and \(y\in{\mathbb {R}}\). By some elementary calculations one can show that
and
for \(s\in[t-r, t]\) and \(y\in{\mathbb {R}}\). Thus, (5.5) follows from Gronwall’s inequality. □
Proof of Theorem 5.2
Let \((t,x)\in[0,T]\times{\mathbb {R}}\). We will adopt a technical argument proposed by Cardon-Weber [29]. By Proposition 2.5, we need only to prove
almost surely. Recall that the statement \(\|Du(t, x)\|_{\mathcal{H}}>0 \) is equivalent to the statement \(\|Du(t,x)\|_{L^{2}([0,T]\times{\mathbb {R}})}>0\). Thus, we need only to introduce \(\|Du(t, x)\|_{L^{2}([0,T ]\times{\mathbb {R}})}> 0\) almost surely. For \(0< r< t\) and \(x\in{\mathbb {R}}\), we denote
and
It follows from (5.1) that
for all \(0< r< t\) and \(x\in{\mathbb {R}}\).
Now, let us to estimate \(\Lambda^{1}(t,x,r)\) and \(\Lambda^{2}(t,x,r)\). Similar to the proof of (5.6), one can see that
By (3.8) and Lemma 5.1, one can also see that
for \(0< r< t\) and \(x\in{\mathbb {R}}\). Combining this with (5.6), (5.7), and (5.8), we get
for all \(r_{0}> 0\), and the theorem follows. □
References
Balan, R, Conus, D: A note on intermittency for the fractional heat equation. Stat. Probab. Lett. 95, 6-14 (2014)
Song, J: On a class of stochastic partial differential equations (2015). arXiv:1503.06525v2
Chen, X, Hu, Y, Song, J: Feynman-Kac formula for fractional heat equations driven by fractional white noises (2012). arXiv:1203.0477
Hu, Y, Nualart, D, Song, J: Feynman-Kac formula for heat equation driven by fractional white noises. Ann. Probab. 39, 291-326 (2011)
Hu, Y, Lu, F, Nualart, D: Feynman-Kac formula for the heat equation driven by fractional noises with Hurst parameter \(H < 1/2\). Ann. Probab. 40, 1041-1068 (2012)
Bo, L, Jiang, Y, Wang, Y: On a class of stochastic Anderson models with fractional noises. Stoch. Anal. Appl. 26, 256-273 (2008)
Diop, M, Huang, J: Retarded neutral stochastic equations driven by multiplicative fractional Brownian motion. Stoch. Anal. Appl. 32, 820-839 (2014)
Duncan, TE, Maslowski, B, Pasik-Duncan, B: Fractional Brownian motion and stochastic equations in Hilbert spaces. Stoch. Dyn. 2, 225-250 (2002)
Balan, R: Recent advances related to SPDEs with fractional noise. Prog. Probab. 67, 3-22 (2013)
Hu, Y, Nualart, D: Stochastic heat equation driven by fractional noise and local time. Probab. Theory Relat. Fields 143, 285-328 (2009)
Liu, J, Yan, L: On a semilinear stochastic partial differential equation with double-parameter fractional noises. Sci. China Math. 57, 855-872 (2014)
Jiang, Y, Wang, X, Wang, Y: On a stochastic heat equation with first order fractional noises and applications to finance. J. Math. Anal. Appl. 396, 656-669 (2012)
Jiang, Y, Wei, T, Zhou, X: Stochastic generalized Burgers equations driven by fractional noises. J. Differ. Equ. 252, 1934-1961 (2012)
Balan, R, Tudor, CA: The stochastic heat equation with a fractional-colored noise: existence of the solution. ALEA Lat. Am. J. Probab. Math. Stat. 4, 57-87 (2008)
Balan, RM, Tudor, CA: Stochastic heat equation with multiplicative fractional-colored noise. J. Theor. Probab. 23, 834-870 (2010)
Tudor, CA: Recent developments on stochastic heat equation with additive fractional-colored noise. Fract. Calc. Appl. Anal. 17, 224-246 (2014)
Sobczyk, K: Stochastic Differential Equations with Applications to Physics and Engineering. Kluwer Academic, London (1991)
Droniou, J, Imbert, C: Fractal first-order partial differential equations. Arch. Ration. Mech. Anal. 182, 299-331 (2006)
Chen, Z, Kumagai, T: Heat kernel estimates for jump processes of mixed types on metric measure spaces. Probab. Theory Relat. Fields 140, 277-317 (2008)
Chen, Z, Kumagai, T: Heat kernel estimates for stable-like processes on d-sets. Stoch. Process. Appl. 108, 27-62 (2003)
Chen, Z, Kim, P, Song, R: Global heat kernel estimate for relativistic stable processes in exterior open sets. J. Funct. Anal. 263, 448-475 (2012)
Chen, Z, Kim, P, Song, R: Heat kernel estimates for \(\Delta ^{\alpha/2}+\Delta ^{\beta /2}\). Ill. J. Math. 54, 1357-1392 (2000)
Chen, Z, Kim, P, Song, R: Heat kernel estimates for \(\Delta +\Delta ^{\alpha/2}\) in \(C^{1,1}\) open sets. J. Lond. Math. Soc. 84, 58-80 (2011)
Nualart, D: Malliavin Calculus and Related Topics. Springer, Berlin (2006)
Wei, T: High-order heat equations driven by multi-parameter fractional noises. Acta Math. Sin. Engl. Ser. 26, 1943-1960 (2010)
Walsh, JB: An introduction to stochastic partial differential equations. In: Ecole d’été de Probabilités de Saint Flour XIV. Lecture Notes in Math., vol. 1180, pp. 266-439. Springer, Berlin (1986)
Dalang, R: Extending the martingale measure stochastic integral with applications to spatially homogeneous SPDE’s. Electron. J. Probab. 4, 1-29 (1999)
Zhang, T, Zheng, W: SPDEs driven by space-time white noises in high dimensions: absolute continuity of the law and convergence of solutions. Stoch. Stoch. Rep. 75, 103-128 (2003)
Cardon-Weber, C: Cahn-Hilliard stochastic equation: existence of the solution and its density. Bernoulli 7, 777-816 (2000)
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The project was sponsored by the National Natural Science Foundation of China (11571071, 71571001) and the Natural Science Foundation of Anhui Provincial (1608085MA02).
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LTY and DFX carried out the mathematical studies, participated in the sequence alignment, drafted the manuscript, and participated in the design of the study and performed proofs of the results. All authors read and approved the final manuscript.
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Xia, D., Yan, L. On a semilinear mixed fractional heat equation driven by fractional Brownian sheet. Bound Value Probl 2017, 7 (2017). https://doi.org/10.1186/s13661-016-0736-y
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DOI: https://doi.org/10.1186/s13661-016-0736-y