Large deviations for excursions of non-homogeneous Markov processes

TL;DR

This paper analyzes the large deviations of ergodic non-homogeneous Markov processes in two dimensions, showing that probabilities of long excursions decay exponentially with a rate proportional to the square of the scaling parameter.

Contribution

It provides a new large deviation principle for non-homogeneous Markov processes with position-dependent jump rates, including explicit rate functions.

Findings

Probabilities of long excursions decay exponentially fast.

The rate function is explicitly calculated for continuous paths.

Long excursions out of zero become exponentially unlikely as scale increases.

Abstract

In this paper, the large deviations on trajectory level for ergodic Markov processes are studied. These processes take values in the non-negative quadrant of the two dimension lattice and are concentrated on step-wise functions. The rates of jumps towards the axes (jump down) depend on the position of the process -- the higher the position, the greater the rate. The rates of jumps going in the same direction as the axes (jump up) are constants. Therefore the processes are ergodic. The large deviations are studied under equal scalings of both space and time. The scaled versions of the processes converge to 0. The main result is that the probabilities of long excursions out of 0 tend to 0 exponentially fast with an exponent proportional to the square of the scaling parameter. A proportionality coefficient is an integral of a linear combination of path components. A rate function of the large deviation principle is calculated for continuous functions only.