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

Abstract

Let u(t,x) be the solution to a stochastic heat equation ∂∂tu=12∂2∂x2u+∂2∂t∂xX(t,x),t≥0,x∈R\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\frac{\\partial }{\\partial t}u=\\frac{1}{2} \\frac{\\partial ^{2}}{\\partial x^{2}}u+ \\frac{\\partial ^{2}}{\\partial t\\,\\partial x}X(t,x),\\quad t\\geq 0, x\\in { \\mathbb{R}} $$\\end{document} with initial condition 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). In fact, it is well known that the solution is a Gaussian process such that the process tmapsto u(t,x) is a bi-fractional Brownian motion with Hurst indices H=K=frac{1}{2} for every real number x. However, the many properties of the process xmapsto u(cdot ,x) are unknown. In this paper we consider the generalized quadratic covariations of the two processes xmapsto u(cdot ,x),tmapsto u(t,cdot ). We show that xmapsto 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.