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May, 1985 On Evaluating the Donsker-Varadhan $I$-Function
Ross Pinsky
Ann. Probab. 13(2): 342-362 (May, 1985). DOI: 10.1214/aop/1176992995


Let $x(t)$ be a Feller process on a complete separable metric space $A$ and consider the occupation measure $L_t(\omega, \cdot) = \int^t_0 \chi_{(\cdot)}(x(s)) ds$. The $I$-function is defined for $\mu \in \mathscr{P}(A)$, the set of probability measures on $A$, by $I(\mu) = -\inf_{u\in\mathscr{D}^+} \int_A (Lu/u)d\mu$ where $(L, \mathscr{D})$ is the generator of the process and $\mathscr{D}^+ \subset \mathscr{D}$ consists of the strictly positive functions in $\mathscr{D}$. The $I$-function determines the asymptotic rate of decay of $P((1/t)L_t(\omega, \cdot) \in G)$ for $G \subset \mathscr{P}(A)$. The first difficulty encountered in evaluating $I(\mu)$ is that the domain $\mathscr{D}$ is generally not known explicitly. In this paper, we prove a theorem which allows us to restrict the calculation of the infimum to a nice subdomain. We then apply this general result to diffusion processes with boundaries.


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Ross Pinsky. "On Evaluating the Donsker-Varadhan $I$-Function." Ann. Probab. 13 (2) 342 - 362, May, 1985.


Published: May, 1985
First available in Project Euclid: 19 April 2007

zbMATH: 0607.60024
MathSciNet: MR781409
Digital Object Identifier: 10.1214/aop/1176992995

Primary: 60F10
Secondary: 60J60

Keywords: diffusion processes with boundaries , large deviations , Martingale problem , occupation measure

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 2 • May, 1985
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