系数奇异且依赖于分布的随机(泛函)微分方程

In the ordinary differential equations, if the coefficients are not Lipschitz continuous (called singular), the solution may not have uniqueness. As to the stochastic differential equations, when the coefficients satisfy some integrable conditions, there is unique strong solution. In order to characterize non-linear path-dependent Fokker-Planck-Kolmogorov equations by probability method, distribution dependent stochastic functional differential equations are introduced. When the coefficients are Lipschitz continuous, distribution dependent stochastic (functional) differential equations have unique strong solution. We will investigate distribution dependent stochastic (functional) differential equations with singular coefficients in this project. We try to combine Zvonkin transform with the existed results for the classical stochastic (functional) differential equations with singular coefficients to prove the existence and uniqueness for the distribution dependent stochastic (functional) differential equations with singular coefficients. Moreover, applying the coupling by change of measure, we expect to obtain the distribution property for the solution, for instance, Harnack and shift Harnack inequalities, gradient estimate and the convergence of the semigroup. Finally, we plan to derive the convergence rate for the numerical method.

在常微分方程中，如果系数不是Lipschitz连续的（称为奇异），方程的解可能不唯一。对于随机微分方程，当系数满足一定的可积性条件，可以证明强解的存在唯一性。为了用概率的方法刻画（路径依赖的）非线性的Fokker-Planck-Kolmogorov 方程，需要引入系数依赖于分布的随机（泛函）微分方程。在系数满足Lipschitz条件下，系数依赖于分布的随机（泛函）微分方程具有唯一强解。本项目探讨系数奇异且依赖于分布的随机（泛函）微分方程，试图结合 Zvonkin 变换以及已有的系数奇异的经典随机（泛函）微分方程的结果在一定条件下证明系数奇异且依赖于分布的随机（泛函）微分方程的解的存在唯一性。更进一步地，通过变测度耦合的方法，试图得到解的分布性质，如 Harnack 和 shift Harnack 不等式、梯度估计、半群的收敛性以及数值算法的收敛速率。