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Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces. (English) Zbl 1391.60049

Author’s abstract: The large deviation principle obtained for the trajectory of empirical distributions of weakly interacting copies of diffusion processes, and similar results for Markov-jump processes, raise the question whether a context independent approach is possible to prove large deviations in the Skorokhod space of \(P(E)\) valued trajectories, for trajectories of weakly interacting, or even independent copies of process on some space \(E\) with a rate function of “Lagrangian” form: \(I(\nu ): = {I_0}(\nu (0)) + \int_0^\infty {L(\nu (s),\dot \nu (s))ds} \) if \(\nu \) is absolutely continuous, and \(I(\nu ): = \infty \) otherwise. The main goal of this paper is to rewrite this contracted rate-function in a unified way applicable for a large class of state spaces and processes. The paper is organized as follows. We start in Section 2 with the preliminaries and state two main results. Theorem 2.1 gives, under the condition that the process solves the martingale problem, the large deviation principle. Under the condition that there exists a suitable core for the generator of the process, Theorem 2.8 gives the decomposition of the rate function. In Section 3, we study functional analytic properties of the generator, its non-linear counterpart \(H\) and the Lagrangian. Additionally, we show that \(H\) is intimately related to the Girsanov transforms of the Markov process. In Section 4, we prove Theorem 2.8. In particular, we introduce the Nisio semigroup in terms of absolutely continuous trajectories and the Lagrangian. In Section 5, we give four examples where Theorem 2.8 applies.

MSC:

60F10 Large deviations
60J25 Continuous-time Markov processes on general state spaces
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