Besov priors for Bayesian inverse problems

We consider the inverse problem of estimating a function $u$ from noisy, possibly nonlinear, observations. We adopt a Bayesian approach to the problem and widen the existing theory, which is developed primarily for Gaussian prior measures, to the case of Besov priors. In doing so a key technical tool, established here, is the development of a Fernique-like theorem for Besov measures. This theorem enables us to identify appropriate conditions on the forward solution operator which imply the well-definedness and well-posedness of the posterior measure, and to match the prior Besov measure to these conditions. We then consider the application of these results to the inverse problem of finding the diffusion coefficient of an elliptic partial differential equation, given noisy measurements of its solution.