On a semilinear mixed fractional heat equation driven by fractional Brownian sheet

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• 作者：Dengfeng Xia ; Litan Yan
• 关键词：stochastic partial differential equations ; fractional Brownian sheet ; Hölder regularity ; density of the law ; Malliavin calculus ; p ; variation
• 刊名：Boundary Value Problems
• 出版年：2017
• 出版时间：December 2017
• 年：2017
• 卷：2017
• 期：1
• 全文大小：1560KB
• 刊物主题：Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general;
• 出版者：Springer International Publishing
• ISSN：1687-2770
• 卷排序：2017

文摘

In this paper, we consider the stochastic heat equation of the form $$\frac{\partial u}{\partial t}=(\Delta+\Delta_{\alpha})u+\frac {\partial f}{\partial x}(t,x,u)+ \frac{\partial^{2}W}{\partial t\, \partial x}, $$ 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.Keywordsstochastic partial differential equationsfractional Brownian sheetHölder regularitydensity of the lawMalliavin calculusp-variationMSC60G2260H0760H151 IntroductionStochastic 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 $$ \left \{ \textstyle\begin{array}{l} \frac{\partial u}{\partial t}=(\Delta+\Delta_{\alpha})u +\frac{\partial f}{\partial x}(t,x,u(t,x))+\frac{\partial ^{2}{W}}{\partial t\, \partial x},\quad t\in[0,T],x\in{\mathbb {R}}, \\ u(0,x)={\vartheta}(x),\quad x\in\mathbb{R} \end{array}\displaystyle \right . $$ (1.1) 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 1For some \(p\geq2\), we have $$ \sup_{x \in\mathbb{R}} \mathbb{E}\bigl(\bigl\vert {\vartheta}(x)\bigr\vert ^{p}\bigr) < +\infty, $$ (1.2) and there is a constant \(\theta \in(0, 1)\) with \(p\theta < 1\) such that $$ \sup_{x \in\mathbb{R}} \mathbb{E}\bigl(\bigl\vert {\vartheta} \bigl(x+x'\bigr)-{\vartheta }(x)\bigr\vert ^{p}\bigr) < C_{p}\bigl\vert x'\bigr\vert ^{p\theta }. $$ (1.3)Assumption 2For each \(T>0\), there exists a constant \(C>0\) such that $$\begin{aligned}& \bigl\vert f(t,x,y)\bigr\vert \leq C\bigl(1+\vert y\vert \bigr), \end{aligned}$$ (1.4)$$\begin{aligned}& \bigl\vert f(t,x,y)-f\bigl(s,x',y'\bigr)\bigr\vert \leq C\bigl(\vert t-s\vert +\bigl\vert x-x'\bigr\vert +\bigl\vert y-y'\bigr\vert \bigr) \end{aligned}$$ (1.5) 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.