Small Perturbation of Diffusions in Inhomogeneous Media

– For the system of d-dim stochastic differential equations, dXε(t)= b(X(t))dt + εσ(X(t))dW(t), t ∈ [0,1], Xε(0)= x0 ∈Rd, where b(x) and σ(x) are smooth except possibly along the hyperplane {(x1, . . . , xd);x1 = 0}, we shall demonstrate that the natural setup of its large deviation principle is to consider the probability ε2 logP(‖Xε − φ‖< δ, ‖uε − ψ‖ < δ, ‖ ε − η‖< δ)∼−I (φ,ψ,η) of the triplet (X,u, ) simultaneously. Here, u is the occupation time of X 1(·) in the positive half line and ε(·) is the local time of X 1(·) at 0. The explicit form of the rate function I (·, ·, ·) is obtained. The usual Wentzell–Friedlin theory concerns only probabilities of the form ε2 logP(‖Xε − φ‖< δ) and its limit is a consequence of the contraction principle of our result.  2002 Éditions scientifiques et médicales Elsevier SAS 1991 MSC: Primary 60J10, Secondary 60J05