Quadratic covariations for the solution to a stochastic heat equation

Abstract

Let $u(t,x)$ be the solution to a stochastic heat equation $$\frac{\partial }{\partial t}u=\frac{1}{2}\frac{{\partial }^2}{\partial x^2}u+\frac{{\partial }^2}{\partial t\partial x}X(t,x),t\ge 0,x\in \mathbb{R}$$ 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 $H=\frac{1}{4}$ 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.

35 pages