不规则对称Lévy型过程的研究

对称Lévy型过程和与其紧密联系的非局部Dirichlet型是随机分析中的热点。在实际应用中，我们常常碰到行走维数依赖空间变量或者系数退化等不规则的对称Lévy型过程，它们在平均场交互作用粒子系统的极限行为、非一致椭圆积分型算子的分析性质等问题中起重要作用。鉴于此，本项目旨在研究不规则对称Lévy型过程，建立行走维数依赖空间变量对称Lévy型过程的热核估计和抛物Harnack不等式，给出系数退化对称Lévy型过程的局部概率和分析性质，同时也将系统探讨长程跳马氏过程的不变原理，并应用于遍历随机环境中相关问题的研究。由于现有研究对称Lévy型过程的工具依赖于行走维数与空间变量的不依赖性及系数的一致椭圆性，不适用于研究不规则对称Lévy型过程的性质，必须建立新的研究方法，因此本项目的研究内容具有挑战性，且拟提出的方法主要基于我们在非局部算子热核估计研究中丰富积累，具有很强的可行性。

Symmetric Lévy type processes and their closely related non-local Dirichlet forms have been intensively studied in the field of stochastic analysis. However, in practical applications, we are often face with irregular symmetric Lévy type processes, including the cases that associated walk dimensions are spatially dependent or associated coefficients are degenerate. Indeed, two typical examples above play important roles in studies of limit behaviors for mean-field interacting particle systems and analytic properties for non-elliptic integral operators. Heavily motivated by these facts, in this proposal we are concerned on irregular symmetric Lévy type processes. We will establish heat kernel estimates and parabolic Harnack inequalities for symmetric Lévy type processes with space-dependent walk dimensions, and gain locally probabilistic and analytic properties for symmetric Lévy type processes with degenerate coefficients. We will also investigate systematically the invariance principle of long range Markov jump processes and, as an application, handle related problems in the ergodic random media. Since the existing tools rely on the space-dependent property of walk dimensions and the uniformly elliptic property of the associated coefficients, they are now invalid for irregular Lévy type processes. Based on this viewpoint, we have to put forward new methods to study irregular symmetric Lévy type processes, so that our proposal is challenging as far as the research topic is concerned. Furthermore, the approaches in this proposal are of high feasibility due to our rich accumulations on heat kernel estimates for non-local operators.