You can watch me talk about this at youtube.com/watch?v=_GJVwmaeH9U (see also maths.dur.ac.uk/lms/106/movies/1341kend.mp4). Briefly: Dirichlet forms can be used to deliver proofs of optimal scaling results for Markov chain Monte Carlo algorithms (specifically, Metropolis-Hastings random walk samplers) under regularity conditions which are substantially weaker than those required by the original approach (based on the use of infinitesimal generators). The Dirichlet form methods have the added advantage of providing an explicit construction of the underlying infinite-dimensional context. In particular, this enables us directly to establish weak convergence to the relevant infinite-dimensional distributions.
I expect that the work discussed in the talk (Zanella, Kendall, Bédard 2016) to be the first-fruits of what may prove to be a very long programme of work, engaging with the application of stochastic analysis to Markov chain Monte Carlo. Indeed, I have recently been awarded an EPSRC grant to work with a research associate on this programme. But there is much to be done, and plenty of room as well as great opportunities here for a PhD student with a good background in mathematical probability and an interest in its applications to computational issues.
1. Zanella, G., Bédard, M., & Kendall, W. S. (2016). A Dirichlet Form approach to MCMC Optimal Scaling. Stochastic Processes and Their Applications 127(12) 4053-4082 (Gold Access). Also arXiv, 1606.01528.