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

Abstract Let u ( t , x ) $u(t,x)$ be the solution to a stochastic heat equation ∂ ∂ t u = 1 2 ∂ 2 ∂ x 2 u + ∂ 2 ∂ t ∂ x X ( t , x ) , t ≥ 0 , x ∈ R $$ \frac{\partial }{\partial t}u=\frac{1}{2} \frac{\partial ^{2}}{\partial x^{2}}u+ \frac{\partial ^{2}}{\partial t\,\partial x}X(t,x),\quad t\geq 0, x\in { \mathbb{R}} $$ with initial condition u ( 0 , x ) ≡ 0 $u(0,x)\equiv 0$ , 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 ↦ u ( t , x ) $t\mapsto u(t,x)$ is a bi-fractional Brownian motion with Hurst indices H = K = 1 2 $H=K=\frac{1}{2}$ for every real number x. However, the many properties of the process x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ are unknown. In this paper we consider the generalized quadratic covariations of the two processes x ↦ u ( ⋅ , x ) , t ↦ u ( t , ⋅ ) $x\mapsto u(\cdot ,x),t\mapsto u(t,\cdot )$ . We show that x ↦ u ( ⋅ , x ) $x\mapsto u(\cdot ,x)$ 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 formulas and Bouleau–Yor identities are introduced.