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

该项目将涉及研究各种类型的马尔可夫过程的收敛，包括随机微分方程解(有跳跃和没有跳跃)和马尔可夫链。我们将证明在一些较温和的假设下，一些关于某些发散的马尔可夫过程具有指数收敛速度。然后，我们将对更一般的McKean-Vlasov随机方程证明类似的结果。这种类型的结果，除了它们的理论意义外，在计算统计和机器学习中也得到了大量的应用。例如，通过使用概率耦合技术，人们可以分析许多蒙特卡罗算法的收敛，这些算法是通过离散随机微分方程而获得的，并用于计算统计中从高维概率分布中进行抽样。

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.