借粒子方法与反射随机微分方程

本项目将研究如下问题：1.Levy过程驱动的反射随机微分方程的解的分布密度的存在性；2.分数次Poisson泛函的局部化及其在密度近似计算上的应用；3.斜反射随机微分方程的解的支集的确定；4.二阶椭圆与抛物方程的Neumann问题的粘性解与分布解的等价性问题；5.反射随机微分方程的解的数值计算问题；6.边界不光滑时二阶椭圆与抛物方程的Neumann问题的解的概率表示；7.对称稳定过程的局部时和自交局部时在Malliavin Calculus意义下的光滑性；8.系数退化时Levy过程驱动的随机延迟微分方程的密度的正则性；9.Levy过程驱动的随机微分方程的解的密度的Varadhan估计; 10.Levy过程驱动的随机微分方程的欧拉折线逼近的弱收敛速度;11.Levy过程驱动的随机延迟微分方程的欧拉折线逼近的弱收敛速度;

In this proposal the following problems will be studied: 1. The existence of the distribution density of the solutions of reflected stochastic differential equations driven by Levy processes; 2. Localization of fractional Poisson functionals and its applications to the approximate calculation of densities; 3. The determination of the support of obliquely reflected stochastic differential equations; 4. The equivalence of viscosity solutions and distribution ones of the Neumann problem of elliptic and parabolic equations of second order; 5. Numerical computation of solutions of reflected stochastic differential equations; 6. Probabilistic representation of the solutions of the Neumann problem with nonsmooth boundary of elliptic and parabolic equations of the second order; 7. Smoothness in the sense of Malliavin calculus of local times and self-intersection local times of symmetric stable processes; 8. Regularity of densities of stochastic delay differential equations with degenerate coefficients; 9. The Varadhan estimate for the density of the solutions of stochastic differential equations driven by Levy processes; 10. The rate of weak convergence of the Euler scheme of stochastic differential equations driven by Levy processes; 11. The rate of weak convergence of the Euler scheme of stochastic delay differential equations driven by Levy processes.