非对称马氏过程的泛函不等式与应用

Functional inequality is a powerful tool in characterizing properties of Markov processes, and a complete theory has been established in the symmetric case. However, many applied models are non-symmetric, for instances, most stochastic (partial) differential equations, stochastic functional differential equations, and degenerate diffusion processes are non-symmetric but fundamental in related fields. Due to unknown invariant probability measures, degenerate noise, or singular coefficients (only with certain integrability condition) , exiting tools developed for symmetric models are no longer valid. Moreover, the energy form in usual functional inequalities is the symmetric part of the Dirichlet form, which is not enough to characterize non-symmetric Markov processes, so that we need to search for new energy form which includes also information from the non-symmetric part. So, it is an important and challenging project to study functional inequalities and applications for non-symmetric Markov processes. On the other hand, due to our solid foundation of functional inequalities and rich experience in the study of non-symmetric, singular and degenerate processes, the project is of high feasibility.

泛函不等式是研究马氏过程的重要研究工具，在对称情形已经形成了完整的理论。然而，在应用中许多数马氏过程是非对称的，比如随机（偏）微分方程、随机泛函微分方程和退化的扩散过程等重要模型，它们大多是非对称的。由于这些模型的平稳分布通常是未知的、噪声是退化的或系数是奇异的（仅具有某种可积性），许多在对称情形发展的研究工具(比如Lyapunov条件)不再适用。通常的泛函不等式中的能量型就是相应Dirichlet型的对称部分，因而当过程非对称时不能完整地刻画过程的性质，需要在能量型中纳入非对称部分的信息，并发展相应于新能量型的分析方法。因此，如何建立非对称马氏过程的泛函不等式并用来刻画过程的概率与分析性质，是一个具有挑战性的重要课题。另一方面，基于我们在泛函不等式方面坚实的研究基础，和近年来在非对称、奇异、退化随机（偏）微分方程研究中的丰富积累，项目具有很强的可行性。

以非对称马氏过程为主线，刻画一些具有重要研究背景和学术价值的随机系统的性质，取得了一批新的研究成果，包括判别奇异的布依-路劲赖随机(偏)微分方程的存在唯一性、建立导数公式、证明遍历性，构造测度值过程并建立泛函不等式研究遍历性、得到随机(偏)微分方程经验测度的Wasserstein极限的精确刻画，研究随机系统的数值算法等， 为相关课题的进一步发展提供新的研究基础和工具。