Existence and regularity of density for solutions to stochastic differential equations with boundary conditions

We study the existence and regularity of the probability law associated with the solution of stochastic differential equations with boundary conditions. We focus our attention on two cases. In the one dimensional case we consider the general sde with a linear boundary condition. In the multidimensional case we consider a general boundary condition with constant diffusion coefficient. Keywords: stochastic differential equations with boundary conditions, density, Malliavin calculus. AMS 1990 SCI: 60H07, 60H10, 60H99 . Supported by a grant of the DGICYT No. PB 930052 y This article was partially written while the author was visiting Kyoto University, Department of Mathematics, under a JSPS fellowship 1 Introduction Stochastic differential equations (sde) with boundary conditions have been studied by Ocone-Pardoux [8], Nualart- Pardoux [10], Donati-Martin [2], Garnier [5], among others. These are equations of the type dX t = oe(X t ) ffi dW t + b(X t ) dt; 0 t 1; (1.1) with a bo...