Almost sure approximation of Wong-Zakai type for stochastic partial differential equations
type: Item
instance of: scholarly article
title: Almost sure approximation of Wong-Zakai type for stochastic partial differential equations
zbMATH Open document ID: 0842.60062
DOI: 10.1016/0304-4149(94)00037-T
author: Zdzislaw Brzezniak
author: Franco Flandoli
published in: Stochastic Processes and their Applications
publication date: 1996-08-01T00:00:00Z
review text: The authors obtain results of almost sure convergence of the solutions of the approximation problems \[ {{\partial u^n (t,x)} \over {\partial t}}+ A u^n (t,x)+ \sum^k_{j=1} B^j u^n (t,x) \dot w^j_n (t)= 0, \qquad u^n (0, x)= u_0 (x), \] to the solution of the problem \[ du (t,x)+ \overline {A} u(t, x)dt+ \sum^k_{j=1} B^j u(t,x) dw^j (t)= 0, \qquad u(0,x)= u_0 (x), \] where \(A\) and \(B^j\) are, respectively, second-order and first-order differential operators, \(\overline {A}= Au+ {1\over 2} \sum^k_{j=1} (B^j)^2\), \(w= (w^1, \dots, w^k)\) is a \(k\)-dimensional Brownian motion, \(w_n\) is the standard piecewise linear approximation of \(w\). The proof of the results is based on employing a generalized Feynman-Kac formula due to Pardoux/Rozovskij (1979, 1983) and on improving the results on the convergence in \(L_p\)-norms of the above approximation procedure (via getting a rate of this convergence) due to the authors and \textit{M. Capiński} [ibid. 8, No. 3, 293-313 (1990; Zbl 0709.60063)]. The main results are applied to the stochastic PDE in the Stratonovich form. The comparison of the results with the results of the previous work of J. M. Bismut (1981) and J. M. Moulinier (1988) are given.
reviewed by: Hong Thai Nguyen
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 60H20
Mathematics Subject Classification ID: 65M99
zbMATH DE Number: 735489
zbMATH Keywords: stochastic partial differential equations
zbMATH Keywords: almost sure convergence
zbMATH Keywords: approximation methods
zbMATH Keywords: Brownian motion
zbMATH Keywords: Feynman-Kac formula
MaRDI profile type: Publication
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full work available at URL: 0304-4149(94)00037-t
OpenAlex ID: W2092317904
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description: scientific article; zbMATH DE number 735489