Relations between stochastic and partial differential equations in hilbert spaces

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 d X (t) = A X (t) d t + B d W (t) and solutions to the deterministic partial differential (with derivatives in Hilbert spaces) equation for the probability characteristic t, x h (X (T)) is proved. Interpretation of objects in the equations is given. © 2012 I. V. Melnikova and V. S. Parfenenkova.