该项目主要研究概率论中的泛函不等式和一类马氏过程的Cutoff现象。泛函不等式包括流形上扩散过程的对数Sobolev不等式与某一维扩散对数Sobolev不等式的比较、L2传输不等式与L1+Poincare不等式之间的等价关系探讨、L2传输不等式与L2传输信息不等式的严格强弱关系、非对称马氏过程的传输信息不等式、一些具体随机（偏）微分方程的泛函不等式、一类有实际意义测度的泛函不等式等。我们也将研究一类马氏过程的cutoff现象，包括生灭过程全变差下cutoff的显示表达和高维空间中一般马氏过程的scutoff问题（扩散过程、Levy过程等）。此cutoff现象的研究涉及到马氏过程击中时问题，特别是强平稳时和最优耦合时间等精确分布问题和过程的Mobius单调性研究。

This project is devoted to functional inequalities and cutoff phenomenon for some Markov processes. For functional inequalities: intend to provide a comparison of logarithmic Sobolev constants respectively for diffusions on manifolds and for corresponding one-dimensional diffusion; investigate whether L1 transportation working together with Poincare inequality imply L2 transportion; L2 transportation information is stricting stronger than L2 transportation?; functional inequalities for some stochastic (partial) differential equations and some .particular proabilities arising from practical model; transportation information inequalities for non symmetric Markov processes. We plan to work on cutoff phenomenon for Markov processes such as explicit criteria on total variation cutoff for birth and death chains and cutoff phenomenon for general multi-dimensional Markov processes, e.g., diffusions and Levy processes. The research on cutoff phenomenon involves: the hitting time of Markov processes, especilly the distribution of the strong stationary time and the optimal coupling time; Mobius stochastic monotonicity of Markov processes.