Bayesian inverse problems for Burgers and Hamilton-Jacobi equations with white-noise forcing

TL;DR

This paper develops a Bayesian framework for inverse problems involving Burgers and Hamilton-Jacobi equations with white-noise forcing, establishing conditions for posterior measure well-posedness and applying the theory to infer the noise from observations.

Contribution

It introduces a rigorous Bayesian approach for inverse problems with white-noise forcing in PDEs, including conditions for posterior measure validity and well-posedness.

Findings

Established conditions for Bayesian posterior measure validity

Applied theory to infer white-noise forcing in PDEs

Demonstrated well-posedness of inverse problems with stochastic forcing

Abstract

The paper formulates Bayesian inverse problems for inference in a topological measure space given noisy observations. Conditions for the validity of the Bayes formula and the well-posedness of the posterior measure are studied. The abstract theory is then applied to Burgers and Hamilton-Jacobi equations on a semi-infinite time interval with forcing functions which are white noise in time. Inference is made on the white noise forcing, assuming the Wiener measure as the prior.