Quadratic covariations for the solution to a stochastic heat equation with space-time white noise

Abstract

AbstractLet $u(t,x)$u(t,x) be the solution to a stochastic heat equation $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$∂∂tu=12∂2∂x2u+∂2∂t∂xX(t,x),t≥0,x∈R with initial condition $u(0,x)\equiv 0$u(0,x)≡0, where Ẋ is a space-time white noise. This paper is an attempt to study stochastic analysis questions of the solution $u(t,x)$u(t,x). In fact, it is well known that the solution is a Gaussian process such that the process $t\mapsto u(t,x)$t↦u(t,x) is a bi-fractional Brownian motion with Hurst indices $H=K=\frac{1}{2}$H=K=12 for every real number x. However, the many properties of the process $x\mapsto u(\cdot ,x)$x↦u(⋅,x) are unknown. In this paper we consider the generalized quadratic covariations of the two processes $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$x↦u(⋅,x),t↦u(t,⋅). We show that $x\mapsto u(\cdot ,x)$x↦u(⋅,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 formulas and Bouleau–Yor identities are introduced.