Necessary and sufficient conditions for path-independence of Girsanov transformation for infinite-dimensional stochastic evolution equations

Abstract

Based on a recent result on linking stochastic differential equations on {ℝ}^d to (finite-dimensional) Burger-KPZ type nonlinear parabolic partial differential equations, we utilize Galerkin type finite-dimensional approximations to characterize the path-independence of the density process of Girsanov transformation for the infinite-dimensional stochastic evolution equations. Our result provides a link of infinite-dimensional semi-linear stochastic differential equations to infinite-dimensional Burgers-KPZ type nonlinear parabolic partial differential equations. As an application, this characterization result is applied to stochastic heat equation in one space dimension over the unit interval.

Keywords

Characterization theorem / Burgers-KPZ type nonlinear equations in infinite dimensions / infinite-dimensional semi-linear stochastic differential equations / Galerkin approximation / Girsanov transformation / stochastic heat equation / path-independence / Fréchet differentiation

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