Small perturbation of diffusions in inhomogeneous media (Q1599986)

Let \(dX^\varepsilon (t)=b(X^\varepsilon (t))dt+\varepsilon \sigma(X^\varepsilon(t))dW(t)\), \(t\in[0,1]\), \(X^\varepsilon(0)=x^0\in R^d,\) be a system of \(d\)-dimensional stochastic differential equations, where \(b(x)\) and \(\sigma (x)\) are smooth except possibly along the hyperplane \(\{(x_1,\dots,x_d); x_1=0\}\). The authors will demonstrate that the natural setup of its large deviation principle is to consider the probability \(\varepsilon^2\log P(\|X^\varepsilon-\varphi\|<\delta, \|u^\varepsilon-\psi\|<\delta, \|l^\varepsilon-\eta\|<\delta)\sim-I(\varphi,\psi,\eta)\) of the triplet \((X^\varepsilon, u^\varepsilon, l^\varepsilon)\) simultaneously. Here, \(u^\varepsilon\) is the occupation time of \(X^\varepsilon_1(\cdot)\) in the positive half line and \(l^\varepsilon(\cdot)\) is the local time of \(X^\varepsilon_1(\cdot)\) at 0. The explicit form of the rate function \(I(\cdot,\cdot,\cdot)\) is obtained. The usual Wentzell-Freidlin theory concerns only probabilities of the form \(\varepsilon^2\log P(\|X^\varepsilon-\varphi\|<\delta)\) and its limit is a consequence of the contraction principle of the authors' result.