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On a class of exponential changes of measure for stochastic PDEs

Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart

TL;DR

This work extends exponential change-of-measure techniques to semilinear SPDEs in infinite dimensions by exploiting a Dynkin-martingale-based $h$-transform within the $ ext{π}$-convergence framework. The authors derive conditions on $h$ (positive, in $ ext{dom}_m(L)$ with bounded $h^{-1}Lh$ and differentiability) that yield a Girsanov-type change, ensuring that under the new measure the process $X$ follows a modified SPDE with drift $Q abla_x ext{log} h$. They also establish a finite-horizon version and develop applications to infinite-dimensional diffusion bridges and guided processes, including posterior conditioning and forced marginals, by tying $h$ to transition densities and tractable reference processes. The framework enables explicit SPDE representations for conditioned or guided dynamics and provides a principled approach to sampling via changed measures, with potential impact on simulation and data-assimilation tasks in high- or infinite-dimensional settings.

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.

On a class of exponential changes of measure for stochastic PDEs

TL;DR

This work extends exponential change-of-measure techniques to semilinear SPDEs in infinite dimensions by exploiting a Dynkin-martingale-based -transform within the -convergence framework. The authors derive conditions on (positive, in with bounded and differentiability) that yield a Girsanov-type change, ensuring that under the new measure the process follows a modified SPDE with drift . They also establish a finite-horizon version and develop applications to infinite-dimensional diffusion bridges and guided processes, including posterior conditioning and forced marginals, by tying to transition densities and tractable reference processes. The framework enables explicit SPDE representations for conditioned or guided dynamics and provides a principled approach to sampling via changed measures, with potential impact on simulation and data-assimilation tasks in high- or infinite-dimensional settings.

Abstract

Given a mild solution to a semilinear stochastic partial differential equation (SPDE), we consider an exponential change of measure based on its infinitesimal generator , defined in the topology of bounded pointwise convergence. The changed measure depends on the choice of a function in the domain of . In our main result, we derive conditions on for which the change of measure is of Girsanov-type. The process under is then shown to be a mild solution to another SPDE with an extra additive drift-term. We illustrate how different choices of impact the law of under 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.
Paper Structure (13 sections, 23 theorems, 151 equations)

This paper contains 13 sections, 23 theorems, 151 equations.

Table of Contents

  1. Introduction
  2. Approach and challenges
  3. Related work and Applications
  4. Outline
  5. Preliminaries
  6. On the stochastic basis
  7. On stochastic evolution equations
  8. On transition semigroups and their generators on spaces of polynomial growth
  9. Main result
  10. The change of measure on a finite time interval
  11. Applications
  12. Conditioned SPDEs
  13. Guided processes

Key Result

Theorem 1.1

Under suitable assumptions on $h \in \mathop{\mathrm{dom}}\nolimits_m(L)$, there exists a unique measure $\mathbb{P}^h$ on $(\Omega, \mathcal{F}, (\mathcal{F}_t)_{t \geq 0})$ that satisfies eq: intro_changeofmeasure. Furthermore, the process is a cylindrical Wiener process with respect to $\mathbb{P}^h$. In particular, $X$ under $\mathbb{P}^h$ solves the SPDE

Theorems & Definitions (53)

  • Theorem 1.1: Informal
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • Definition 2.5: $\pi$-semigroup
  • Remark 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • Definition 2.9
  • ...and 43 more