On a class of exponential changes of measure for stochastic PDEs

TL;DR

This paper develops a framework for exponential measure changes in stochastic PDEs, extending Girsanov's theorem to infinite dimensions and illustrating applications like diffusion bridges and guided processes.

Contribution

It introduces conditions for measure changes via the generator of SPDEs, generalizing finite-dimensional Girsanov results to infinite-dimensional stochastic PDEs.

Findings

Derived conditions for Girsanov-type measure changes in SPDEs

Constructed infinite-dimensional diffusion bridges

Introduced guided processes for SPDEs

Abstract

Given a mild solution $X$ to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator $L$, defined in the topology of bounded pointwise convergence. The changed measure $\mathbb{P}^h$ depends on the choice of a function $h$ in the domain of $L$. In our main result, we derive conditions on $h$ for which the change of measure is of Girsanov-type. The process $X$ under $\mathbb{P}^h$ is then shown to be a mild solution to another SPDE with an extra additive drift-term. We illustrate how different choices of $h$ impact the law of $X$ under $\mathbb{P}^h$ in selected applications. These include the derivation of an infinite-dimensional diffusion bridge as well as the introduction of guided processes for SPDEs, generalizing results known for finite-dimensional diffusion processes to the infinite-dimensional case.