Large Deviations for Processes in Random Environments with Jumps

TL;DR

This paper investigates large deviation probabilities for non-Markovian deterministic walks in random environments with jumps, extending classical results to processes with finite-range dependence and discontinuities.

Contribution

It introduces a framework for analyzing large deviations in processes with jumps and non-Markovian dependence, including both discrete and continuous models.

Findings

Exponential decay rates for hitting distant sites in deterministic walks.

Large deviation principles for solutions to ODEs with random coefficients and jumps.

Extension of large deviation theory to non-Markovian, jump processes.

Abstract

A deterministic walk in a random environment can be understood as a general random process with finite-range dependence that starts repeating a loop once it reaches a site it has visited before. Such process lacks the Markov property. We study the exponential decay of the probabilities that the walk will reach sites located far away from the origin. We also study a similar problem for the continuous analogue: the process that is a solution to an ODE with random coefficients. In this second model the environment also has "teleports" which are the regions from where the process can make discontinuous jumps.