Large deviations of the trajectory of empirical distributions of Feller processes on locally compact spaces
| Autor: | Richard C. Kraaij |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2018 |
| Předmět: |
Statistics and Probability
Pure mathematics large deviation principle 60F10 60J25 01 natural sciences Legendre transformation 010104 statistics & probability symbols.namesake 60J25 FOS: Mathematics Doob transform Locally compact space Nisio semigroup 0101 mathematics Lagrangian Mathematics Markov processes Probability (math.PR) 010102 general mathematics Absolute continuity Feller process Functional Analysis (math.FA) Mathematics - Functional Analysis Metric space Bounded function symbols Large deviations theory Statistics Probability and Uncertainty Rate function Mathematics - Probability 60F10 |
| Zdroj: | Ann. Probab. 46, no. 2 (2018), 775-828 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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