Journal of Applied Mathematics and Stochastic Analysis
Volume 2005 (2005), Issue 2, Pages 167-173
doi:10.1155/JAMSA.2005.167
On some stochastic parabolic differential equations in a Hilbert space
Department of Mathematics, Faculty of Science, Alexandria University, P.O. Box 21511, Alexandria, Egypt
Received 12 March 2004; Revised 29 July 2004
Copyright © 2005 Khairia El-Said El-Nadi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider some stochastic difference partial differential equations of the form
du(x,t,c)=L(x,t,D)u(x,t,c)dt+M(x,t,D)u(x,t−a,c)dw(t), where
L(x,t,D) is a linear uniformly elliptic partial differential operator of the second order,
M(x,t,D) is a linear partial differential operator of the first order, and w(t) is a Weiner process. The existence and uniqueness
of the solution of suitable mixed problems are studied for the
considered equation. Some properties are also studied. A more
general stochastic problem is considered in a Hilbert space and
the results concerning stochastic partial differential equations
are obtained as applications.