带跳SDE及其在Boltzmann方程研究中的应用

随机微分方程是概率论中重要的研究课题，它在物理，化学，生物，经济，量子场论等领域有着广泛的应用。基于布朗运动的随机微分方程理论已经得到了很好的发展，而带跳随机微分方程因其广泛适用性成为近些年研究的热点，其非连续性增加了研究的难度。本项目致力于研究Levy驱动的SDE的适定性以及与Boltzmann相关的随机粒子系统收敛性。主要包括以下几个问题：（1）一大类Levy过程驱动的SDE强解的存在唯一性；（2）相对软势情形下，Boltzmann方程弱测度解的唯一性；（3）相对软势情形下，随机Nanbu粒子系统的收敛性； 及（4） 相对软势情形下，随机Kac粒子系统的收敛性。

Stochastic differential equation is one of important hot topic in Probability, it is widely applied in physics, chemistry, biology, economy and quantum fields. The theory of SDE driven by Brownian motion has been well developed. However, the jumping SDE is very popular in recent several decades due to its wide application, and its discontinuity increases the research difficulties. This project is to study: (1) The existence and uniqueness of strong solution to the SDE driven by a large class of Levy processes; (2) The uniqueness of weak measure solution for the Boltzmann equation with moderately soft potentials; (3) The rate of convergence of Nanbu's particle system for the Boltzmann equation with moderately soft potentials; and (4) The rate of convergence of Kac's particle system for the Boltzmann equation with moderately soft potentials.