On the relation between the Girsanov transform and the Kolmogorov equations for SPDEs

TL;DR

This paper explores the connection between the Girsanov transform and Kolmogorov equations in SPDEs, showing their series expansions coincide and extending well-posedness theory.

Contribution

It demonstrates the equivalence of series expansions from Girsanov transform and Kolmogorov equations, and extends well-posedness results beyond bounded nonlinearities.

Findings

Series expansions from Girsanov and Kolmogorov methods coincide.

Extended well-posedness of Kolmogorov equations beyond bounded nonlinearities.

Applied iteration approach to SPDEs for broader conditions.

Abstract

The Girsanov transform and Kolmogorov equations are two useful methods for studying SPDEs. It is shown that, under suitable conditions, the series expansion obtained from the Girsanov transform coincides with the one generated by an iteration scheme for Kolmogorov equations. We also apply the iteration approach to extend the well posedness theory for Kolmogorov equations beyond the boundedness condition on the nonlinear term.