Stochastic heat equation with random coefficients
type: Item
instance of: scholarly article
title: Stochastic heat equation with random coefficients
zbMATH Open document ID: 0939.60065
author: Elisa Alòs
author: David Nualart
author: Jorge A. Leon
publication date: 2000-07-05T00:00:00Z
review text: Given a two-parameter Wiener process \(W=\{W(t,x), (t,x)\in [0,T] \times[0,1]\}\), the authors prove the existence of a unique adapted solution of the one-dimensional stochastic parabolic partial differential equation (SPDE) \[ \begin{aligned}\partial_tu= & {\mathcal A}_tu+ f\bigl(t,x,u(t,x)\bigr)+ g\bigl(t,x,u(t,x) \bigr)\partial^2_{t,x}W(t,x), \quad (t,x)\in [0,T]\times[0,1],\\ & u(t,0)= u(t, 1)=0, \quad u(0,x)= u_0(x),\end{aligned} \] with \({\mathcal A}_t=a(t,x) \partial^2_{xx}+ b(t,x) \partial_x\), where \(a(t,x)\), \(b(t,x)\), \(f(t,x,r)\), \(g(t,x,r)\) are predictable processes, \(f,g\) are globally Lipschitz and of at most linear growth in \(r\). In their approach the authors use the Green kernel \(\Gamma=\{\Gamma_{t, s} (x,y),\;0\leq s\leq t\leq T,\;x,y\in[0,1]\}\), to rewrite the SPDE into an evolution equation containing the stochastic integral \(\int^t_0 \int^1_0 \Gamma_{t,0}(s,y) f(s,y,u(s,y))dW^-(s,y)\). Since \(\Gamma_{t,s}(s,y)\) depends on \(W(s,z)\), \((s,z) \in[0,t] \times[0,1]\), the integrand anticipates \(W\). As shown by the authors, the choice of \(dW^-(s,y)\) as anticipating forward integral [see \textit{F. Russo} and \textit{P. Vallois}, Probab. Theory Relat. Fields 97, No. 3, 403-421 (1993; Zbl 0792.60046)] guarantees equivalence between the SPDE and the evolution equation. To solve the evolution equation, maximal estimates of the \(L^p\)-norms are needed. To obtain them the forward integral is decomposed into the anticipating Skorokhod integral which is treated by means of the Malliavin calculus [see \textit{D. Nualart} and \textit{E. Pardoux}, ibid. 78, No. 4, 535-581 (1988; Zbl 0629.60061)], and into a correcting Lebesgue integral.
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H07
zbMATH DE Number: 1371487
zbMATH Keywords: stochastic partial differential equation
zbMATH Keywords: forward integral
zbMATH Keywords: Skorokhod integral
zbMATH Keywords: Malliavin calculus
reviewed by: Rainer Buckdahn
MaRDI profile type: Publication
full work available at URL: s004400050236
OpenAlex ID: W2061909553
DOI: 10.1007/S004400050236
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published in: Zeitschrift für Wahrscheinlichkeitstheorie und Verwandte Gebiete
description: scientific article; zbMATH DE number 1371487