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March 2018 Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces
Richard C. Kraaij
Ann. Probab. 46(2): 775-828 (March 2018). DOI: 10.1214/17-AOP1192

Abstract

We study the large deviation behaviour of the trajectories of empirical distributions of independent copies of time-homogeneous Feller processes on locally compact metric spaces. Under the condition that we can find a suitable core for the generator of the Feller process, we are able to define a notion of absolutely continuous trajectories of measures in terms of some topology on this core. Also, we define a Hamiltonian in terms of the linear generator and a Lagrangian as its Legendre transform.

We prove the large deviation principle and show that the rate function can be decomposed as a rate function for the initial time and an integral over the Lagrangian, finite only for absolutely continuous trajectories of measures.

We apply this result for diffusion and Lévy processes on $\mathbb{R}^{d}$, for pure jump processes with bounded jump kernel on arbitrary locally compact spaces and for discrete interacting particle systems. For diffusion processes, the theorem partly extends the Dawson and Gärtner theorem for noninteracting copies in the sense that it only holds for time-homogeneous processes, but on the other hand it holds for processes with degenerate diffusion matrix.

Citation

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Richard C. Kraaij. "Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces." Ann. Probab. 46 (2) 775 - 828, March 2018. https://doi.org/10.1214/17-AOP1192

Information

Received: 1 March 2016; Revised: 1 November 2016; Published: March 2018
First available in Project Euclid: 9 March 2018

zbMATH: 06864073
MathSciNet: MR3773374
Digital Object Identifier: 10.1214/17-AOP1192

Subjects:
Primary: 60F10, 60J25

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 2 • March 2018
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