The Generalization Of The Dirichlet Problem For Diffusion Equation

Let D be a bounded domain in the d-fl-dimensional Euclidean space Rd. This paper aims at giving a probabilistic treatment of the Dirichlet problem for the following diffusion equation on D.where x(t) is the position of SSTBM path starting at i(0) at time t, q is a bounded Holder continuous function . The existence and uniqueness of the solution of the stochastic Dirichlet problem for the diffusion equation in 83 is obtained. Furthermore a definition of the Dirichlet problem with vibrational boundary for the diffusion equation is given. At the same time the existence , uniqueness and stabilization of the solution is proved.