Simulation of infinite-dimensional diffusion bridges
Thorben Pieper-Sethmacher, Frank van der Meulen, Aad van der Vaart
TL;DR
This work develops a principled method to sample infinite-dimensional diffusion bridges for semilinear SPDEs conditioned on a finite-dimensional observation $L X_T = y$. By leveraging an exponential change of measure with a Doob $h$-transform, the authors construct a tractable guided process $X^{\circ}$ whose law is absolutely continuous w.r.t. the true bridge $X^{\star}$, enabling efficient Metropolis-Hastings and related inference techniques. They prove limit-behavior and absolute-continuity results, provide explicit constructions using the OU process, and demonstrate the approach on stochastic reaction-diffusion equations, including diagonalisable and neural-field models, with numerical illustrations and practical MH algorithms. The framework advances sampling and smoothing for SPDE bridges and has potential applications in partially observed SPDE parameter estimation and state estimation. Overall, the paper extends finite-dimensional diffusion-bridge methods to the SPDE setting and offers a versatile toolkit for inference in infinite-dimensional stochastic systems.
Abstract
Let X be the mild solution to a semilinear stochastic partial differential equation. In this article, we develop methodology to sample from the infinite-dimensional diffusion bridge that arises from conditioning X on a linear transformation LXT of the final state XT at some time T > 0. This solves a problem that has so far not been attended to in the literature. Our main contribution is the derivation of a path measure that is absolutely continuous with respect to the path measure of the infinite-dimensional diffusion bridge. This lifts previously known results for stochastic ordinary differential equations to the setting of infinite-dimensional diffusions and stochastic partial differential equations. We demonstrate our findings through numerical experiments on stochastic reaction-diffusion equations.