Convergence rates for Markov process with applications to computational statistics and machine learning.

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The project will involve investigating the convergence of various types of Markov processes, including solutions of stochastic differential equations (with and without jumps) and Markov chains. We are going to prove that under some mild assumptions, some Markov processes with respect to some divergences have an exponential rate of convergence. Then, we will prove similar results for McKean-Vlasov stochastic equation which is more general. Results of such type, besides their theoretical significance, have found numerous applications in computational statistics and machine learning. For instance, by employing the probabilistic coupling technique, one can analyse the convergence of numerous Monte Carlo algorithms that are obtained via discretizations of stochastic differential equations and are used in computational statistics for sampling from high dimensional probability distributions.