Dirichlet Boundary Problem For A Class Of Operator And Random Periodic Solution SDE

In this paper,we discuss some topics on the Dirichlet boundary problem for ?+?? and random periodic solution for SDE.First,we apply time reversal of Brownain motion and theory of Dirichlet forms to consider the Dirichlet boundary problem for ?+??.We show that there exists a unique,continuous probabilistic solution.When considering this,first,we consider the function p which is continuous,infinitely differen-tiable and has compact supporting in region D;next,for general ?,we construct a sequence of ?n which are continuous,infinitely differentiable and has compact supporting in region D and ?n converge to p with respect to the norm in Sobolev Space;last,we prove that the probabilistic solution is continuous in boundary of D.Second,we generalize Euler methods to stochastic theta methods.When considering this,first,we make use of stochastic theta methods to construct numerical solution for a class of SDE and prove that this solution is convergent in L2(?)for any initial value ? which satisfies the some condition and satisfies stochastic periodic property;last,we prove that the error between the exact random periodic solution and the approximated one is at the 1/4 order of time step in the mean sense.