Large deviations for continuous additive functionals of symmetric Markov processes
The paper under review studies large deviations for continuous additive functionals of symmetric Markov processes associated with a regular symmetric Dirichlet form. \par The main objective is to extend the results in [\textit{B. Remillard}, in: Stochastic models. Proceedings of the international conference, in honour of Prof. Donald A. Dawson, Ottawa, Canada, June 10--13, 1998. Providence, RI: AMS, American Mathematical Society for the Canadian Mathematical Society. 375--398 (2000; Zbl 0965.60039)], [\textit{M. Takeda}, Potential Anal. 19, No. 1, 51--67 (2003; Zbl 1018.60029)] and [\textit{M. Takeda} and \textit{K. Tsuchida}, Trans. Am. Math. Soc. 359, No. 8, 4031--4054 (2007; Zbl 1112.60058)] to more general symmetric Markov processes. \par Let $X$ be a locally compact separable metric space and $m$ a positive Radon measure on $X$ with full support. Let $\mathbf{M}=(P_x,X_t)$ be an $m$-symmetric Markov process on $X$, and the corresponding Dirichlet form on $L^2(X;m)$ be $(\mathcal{E},D(\mathcal{E}))$. Let $\mu$ be a positive Radon measure in the \textit{Green-tight Kato class} and $A_t^{\mu}$ the positive continuous additive functional in the Revuz correspondence to $\mu$. Under some conditions (e.g., irreducibility, conservativeness, doubly Feller property and local ultra-contractivity), the author obtains the large deviation principle for $A_t^{\mu}$. Finally, a 1-dimensional Brownian motion with positive drift is given as an example.
scientific article; zbMATH DE number 7040979