arXiv 1602.087962016-02math.PR

随机heat equation解的二次协变分

Quadratic covariations for the solution to a stochastic heat equation

Xichao Sun, Litan Yan, Xianye Yu

备注：: 35 pages

摘要：

设$u(t,x)$为随机热方程$$ \frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in {\mathbb R} $$的解，初始条件为$u(0,x)\equiv 0$，其中$X$为时空白噪声.本文是研究解$u(t,x)$的随机分析问题的一个尝试。实际上，该解是一个高斯过程，使得过程$t\mapsto u(t,\cdot)$是一个双分数布朗运动，对每个实数$x$，该过程似乎是一个Hurst指数为$H=\frac14$的分数布朗运动.然而，工艺$x\mapsto u(\cdot,x)$的性质是未知的。本文考虑了两个过程$x\mapsto u(\cdot,x),t\mapsto u(t,\cdot)$的二次协变。证明了$x\mapsto u(\cdot,x)$存在一个非平凡的有限二次变分，且某些适应过程关于它的前向积分与伊藤积分一致，但它不是半鞅.此外，还引入了一些推广的伊藤公式和Bouleau-Yor恒等式.

英文摘要：

Let $u(t,x)$ be the solution to a stochastic heat equation $$ \frac{\partial}{\partial t}u=\frac12\frac{\partial^2}{\partial x^2}u+\frac{\partial^2}{\partial t\partial x}X(t,x),\quad t\geq 0, x\in {\mathbb R} $$ with initial condition $u(0,x)\equiv 0$, where $X$ is a time-space white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$. In fact, the solution is a Gaussian process such that the process $t\mapsto u(t,\cdot)$ is a bi-fractional Brownian motion seemed a fractional Brownian motion with Hurst index $H=\frac14$ for every real number $x$. However, the properties of the process $x\mapsto u(\cdot,x)$ are unknown. In this paper we consider the quadratic covariations of the two processes $x\mapsto u(\cdot,x),t\mapsto u(t,\cdot)$. We show that $x\mapsto u(\cdot,x)$ admits a nontrivial finite quadratic variation and the forward integral of some adapted processes with respect to it coincides with "Itô's integral", but it is not a semimartingale. Moreover, some generalized Itô's formulas and Bouleau-Yor identities are introduced.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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