2014/01/31 by Richard C. Kraaij · 1 citation
Mathematics · #Bounded function #Compact space #Large deviations theory #Markov Chains and Monte Carlo Methods #Mathematical analysis #Mathematics #Random Matrices and Applications #Rate function #Stochastic processes and statistical mechanics #math.FA #math.PR #msc:60F10 #msc:60J25
paper · pdf · doi:10.1214/17-aop1192
published as Annals of Probability 2018, Vol. 46, No. 2, 775-828
arxiv created 2016/11/21 · openalex publication_date 2018/03/01 · arxiv updated 2018/03/13 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 ℝd, 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.