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Zbl 1002.60070
Chiang, Tzuu-Shuh; Sheu, Shuenn-Jyi
Small perturbation of diffusions in inhomogeneous media. (English)
[J] Ann. Inst. Henri Poincaré, Probab. Stat. 38, No.3, 285-318 (2002). ISSN 0246-0203

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.
[Nicko G.Gamkrelidze (Moskva)]

MSC 2000:: *60J10 Markov chains with discrete parameter
60J05 Markov processes with discrete parameter

Keywords: Wentzell-Freidlin theory; large deviation principle

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