Backward Stochastic Differential Evolutionary Systems And Their Applications

The present dissertation is concerned with backward stochastic differential evolutionary systems and their applications.Chapter1is concerned with general backward doubly stochastic differential evolutionary systems. An Ito formula is established for the Banach space-valued backward doubly stochastic differential evolutionary systems. Using the weak convergence method on basis of monotone operator theory, the existence and uniqueness of the solution is proved. As an application, the existence and uniqueness is studied on the weak solution for quasilinear backward doubly stochastic PDEs, which will be used in Chapter4.In Chapter2, the connections are addressed between a class of coupled forward-backward stochastic differential systems and the viscous incompressible Navier-Stokes equations, which generalizes the Feynman-Kac formula and gives a probabilistic representation for the solutions of Navier-Stokes equations. Moreover, a self-contained probabilistic proof is given for the existence and uniqueness of the local Hm-solution, and the local solution can be extended uniquely to be a global one for both cases of dimension2and small Reynolds number.In Chapter3, the LP theory and a comparison theorem are established for the semilinear super-parabolic backward stochastic PDEs on the whole space.In Chapter4, the existence and uniqueness is first shown for the weak solution for the Dirichlet problem of the quasilinear backward stochastic PDEs with Sobolev coefficients on bounded domains. Then the De Giorgi iteration scheme is used to prove the maximum princi-ples and the local maximum estimates.Finally in Chapter5,2-dimensional backward stochastic Navier-Stokes equation with nonlinear external forcing is investigated. The existence and uniqueness is shown for the strong solution under the spatially periodic boundary condition.