带跳随机微分方程的若干问题研究

Stochastic analysis of Lévy process has attached more and more attentions in the last decade. Stochastic differential equation with Lévy noise has been widely used in applications such as biography, physics, economics and controls. Meanwhile, it plays an important role in the development of mathematic theories.. This project aims to study two kinds of singular stochastic differential equations with jumps: stochastic differential equations driven by multiplicative pure jump Lévy process and stochastic differential equations driven by general Lévy noise which mix Brown motion and pure jump Lévy process. We shall focus on the pathwise uniqueness, regularities with respect to the initial point, Malliavin differentiability, strong Feller and irreducible of the strong solution to the above two kinds of stochastic differential equations with irregular coefficients. In the mixing noise case, we shall focus on the relationship between the two kinds of noises. Moreover, we shall also study the long time bahaviour of the unique strong solution, such as the existence of invariant measures, ergodicity as well as the large derivation.

最近十几年里，关于Lévy过程随机分析的研究受到了越来越多学者的重视。由Lévy过程驱动的随机微分方程不仅在生物、物理、金融及控制等各种实际问题中有广泛应用，同时也在数学自身理论的发展中占有重要地位。. 本项目考虑两类具有奇异系数的带跳随机微分方程：由可乘纯跳Lévy过程驱动的随机微分方程和由可乘Brown运动及纯跳Lévy过程混合驱动的随机微分方程。主要研究不规则系数下强解的存在唯一性、强解关于初值的正则性、Malliavin可微性以及对应半群的强Feller性和不可约性等。特别的，对于混合噪音情形，我们将深入研究Brown噪音与纯跳Lévy噪音在稳定系统时的相互影响，寻找相互作用机制。并且，我们还将进一步研究奇异系数下这两类带跳随机微分方程强解的长时间渐近行为，包括不变测度的存在性、遍历性及大偏差原理等。

. 本项目考虑两类具有奇异系数的带跳随机微分方程：由纯跳Lévy过程驱动的随机微分方程和由Brown运动及纯跳Lévy过程混合驱动的随机微分方程。我们证明了不规则系数下带跳随机微分方程强解的存在唯一性，同时研究了强解关于初值的正则性、对应半群的强Feller性、不可约性以及强解的长时间渐进行为。极大地推广了随机微分方程的经典结果。.. 受本项目资助，共发表SCI论文12篇，其中包括Ann. Probab. 1篇，Ann. Inst. Henri Poincare-Pr.、SPA、JDE共5篇。