International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 858736, 9 pages
http://dx.doi.org/10.1155/2012/858736
Research Article

Relations between Stochastic and Partial Differential Equations in Hilbert Spaces

I. V. Melnikova and V. S. Parfenenkova

Institute of Mathematics and Computer Sciences (IMCS), Ural Federal University, Lenin Avenue 51, 620083 Ekaterinburg, Russia

Received 31 May 2012; Accepted 20 August 2012

Academic Editor: Yaozhong Hu

Copyright © 2012 I. V. Melnikova and V. S. Parfenenkova. 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

The aim of the paper is to introduce a generalization of the Feynman-Kac theorem in Hilbert spaces. Connection between solutions to the abstract stochastic differential equation and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic is proved. Interpretation of objects in the equations is given.