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

该项目将涉及调查各种类型的马尔可夫过程的收敛性，包括随机微分方程（有和没有跳跃）和马尔可夫链的解决方案。我们将证明在一些温和的假设下，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，我们将证明类似的结果McKean-Vlasov随机方程，这是更一般的。这种类型的结果，除了它们的理论意义，在计算统计和机器学习中有许多应用。例如，通过采用概率耦合技术，可以分析通过随机微分方程的离散化获得的许多Monte Carlo算法的收敛性，并用于从高维概率分布中进行抽样的计算统计。

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.