Relations between Stochastic and Partial Differential Equations in Hilbert Spaces

V. S. Parfenenkova,I. V. Melnikova

doi:10.1155/2012/858736

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.

Highlights

The Feynman-Kac theorem in the numerical vector case relates solutions of the Cauchy problem for stochastic equations with a Brownian motion W t, t ≥ 0: dX t β t, X t dt γ t, X t dW t, t ∈ 0, T, X 0 y, 1.1 with solutions of the Cauchy problem for deterministic partial differential equations: gt t, x β t, x gx t, x
Et,x means the mathematical expectation of a solution to 1.1 with initial value X t x, 0 ≤ t ≤ T
The process X t describes the random motion of International Journal of Stochastic Analysis particles in a liquid or gas, and g t, x is a probability characteristic such as temperature, determined by the Kolmogorov equation

Summary

Introduction

The Feynman-Kac theorem in the numerical vector case relates solutions of the Cauchy problem for stochastic equations with a Brownian motion W t , t ≥ 0: dX t β t, X t dt γ t, X t dW t , t ∈ 0, T , X 0 y, 1.1 with solutions of the Cauchy problem for deterministic partial differential equations: gt t, x β t, x gx t, x. We prove the infinite-dimensional case of the Feynman-Kac theorem in a standard basic conditions such as A is the generator of a C0-semigroup in a Hilbert space H, B is a bounded operator from a Hilbert space U to H, and W is a U-valued Q-Wiener process. This is done for the clarity of proof despite the fact that the conditions are likely to be weakened, and the statement is true in more general assumptions. Particular attention is paid to the subtle issue of transition from zero expectation for a function of g to equality for g itself

Definitions and Auxiliary Statements

Proof of the Main Result