In Bayesian inverse problems one is interested in performing computations with respect to an infinite dimensional probability distribution. A modern computational approach consists of approximating this infinite dimensional probability distribution by running a truncated version of a genuine infinite dimensional Markov chain. If a well-posed infinite dimensional chain exists, then the truncated, finite-dimensional approximation may be expected to have desirable scaling properties with respect to dimension.

In this talk we present some explorations of this topic in conjunction with the recent advance of Piecewise Deterministic Monte Carlo methods such as the Bouncy Particle Sampler and the Zig-Zag Sampler and more recently the Boomerang Sampler. In particular we provide a general construction of Piecewise Deterministic Markov Processes with unbounded event rates in infinite dimensional spaces, we study the wellposedness of infinite dimensional versons of the the Zig-Zag Sampler, Bouncy Particle Sampler, and the Boomerang Sampler, and for the Boomerang Sampler we provide exponential ergodicity result by means of hypocoercivity techniques.

This is joint work with Paul Dobson.