Almost sure approximation of Wong-Zakai type for stochastic partial differential equations
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.
stochastic partial differential equations
scientific article; zbMATH DE number 735489
scientific article