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.

Key words： 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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