Large deviation of diffusion processes with discontinuous drift

For the system of d-dim stochastic differential equations {dX(epsilon) (t) = b(X-epsilon (t)) dt + epsilon dW (t), t is an element of [0, 1] X-epsilon (0) = x(0) is an element of R-d where b is smooth except possibly along the hyperplane x(1) = 0, we shall consider the large deviation principle for the law of the solution diffusion process and its occupation time as epsilon --> 0. In other words, we consider P(parallel toX(epsilon) - phi parallel to < <delta>, parallel tou(epsilon) - psi parallel to < <delta>) where u(epsilon) (t) and psi (t) are the occupation times of X-epsilon and phi in the positive half space {x is an element of R-d : x(1) > 0) respectively. As a consequence, a unified approach of the lower-level large deviation principle for the law of X-epsilon(.) P(parallel toX(epsilon) - phi parallel to < <delta>) can be obtained.