Simulation of infinite-dimensional diffusion bridges

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.