Small perturbation of diffusions in inhomogeneous media

For the system of d-dim stochastic differential equations, dX(epsilon) (t) = b(X-epsilon(t)) dt + epsilonsigma(X-epsilon(t)) dW(t), t is an element of [0, 1], X-epsilon(0) = x(0) is an element of R-d, where b(x) and sigma (x) are smooth except possibly along the hyperplane {(x(1),..., x(d)): x(1) = 0}, we shall demonstrate that the natural setup of its large deviation principle is to consider the probability epsilon(2) log P(parallel to X-epsilon - phi parallel to < delta. parallel to u(epsilon) - psi parallel to < delta, parallel to l(epsilon) - eta parallel to < delta) similar to -I ( phi, psi, eta) of the triplet (X-epsilon, u(epsilon), l(epsilon)) simultaneously. Here, u(epsilon) is the occupation time of X-1(epsilon)((.)) in the positive half line and l(epsilon)((.)) is the local time of X-1(epsilon)((.)) at 0. The explicit form of the rate function I((.), (.), (.)) is obtained. The usual Wentzell-Friedlin theory concerns only probabilities of the form epsilon(2) log P(parallel to X-epsilon - phi parallel to < delta) and its limit is a consequence of the contraction principle of our result. (C) 2002 tditions scientifiques et medicales Elsevier SAS.