Regularity Of Distribution-dependent Stochastic Differential Equations

This thesis mainly studies the following two aspects:(1)Regularity of solutions of DDSDEs with jumps.Firstly,by using Lions derivative with respect to probability measures,the differentiability of the solution concerning to the initial value as well as the associated estimates are investigated.Then,a Bismut derivative,which is an important tool to study the regularities of semigroups of diffusions,is established by Malliavin calculus.Finally,dimension-free Harnack inequality of the solutions to DDSDEs with additive noise is derived by combing Bismut formula and Young's inequality.Harnack inequality is widely applied to heat kernel estimates,gradient estimates,ergodicity and many other fields.(2)By Malliavin calculus and Lions derivatives,a Bismut formula is established for infinite-dimensional DDSDEs.This generalizes the corresponding results in finite-dimensional case.