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Large deviations for intersection measures of some Markov processes
Author(s) -
Mori Takahiro
Publication year - 2020
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.201800228
Subject(s) - mathematics , measure (data warehouse) , intersection (aeronautics) , rate function , heat kernel , large deviations theory , bounded function , bounded variation , mathematical analysis , probability measure , brownian motion , hausdorff dimension , hausdorff measure , moment (physics) , statistics , physics , classical mechanics , database , computer science , engineering , aerospace engineering
Consider an intersection measure ℓ t IS of p independent (possibly different) m ‐symmetric Hunt processes up to time t in a metric measure space E with a Radon measure m . We derive a Donsker–Varadhan type large deviation principle for the normalized intersection measuret − pℓ t ISon the set of finite measures on E as t → ∞ , under the condition that t is smaller than life times of all processes. This extends earlier work by W. König and C. Mukherjee [16], in which the large deviation principle was established for the intersection measure of p independent N ‐dimensional Brownian motions before exiting some bounded open set D ⊂ R N . We also obtain the asymptotic behavior of logarithmic moment generating function, which is related to the results of X. Chen and J. Rosen [7] on the intersection measure of independent Brownian motions or stable processes. Our results rely on assumptions about the heat kernels and the 1‐order resolvents of the processes, hence include rich examples. For example, the assumptions hold for p ∈ Z with 2 ≤ p < p ∗when the processes enjoy (sub‐)Gaussian type or jump type heat kernel estimates, where p ∗ is determined by the Hausdorff dimension of E and the so‐called walk dimensions of the processes.