Joint density for the local times of continuous-time Markov chains: Extended version

TL;DR

This paper derives an explicit joint density formula for local times of continuous-time Markov chains, enabling advanced large deviation estimates and connecting to classical Ray-Knight theorems.

Contribution

It introduces a novel explicit density formula for local times of Markov chains, facilitating new large deviation bounds and linking to Ray-Knight theorems.

Findings

Derived explicit joint density formula for local times

Established large deviation upper bounds for local times

Connected density formula to Ray-Knight theorem

Abstract

We investigate the local times of a continuous-time Markov chain on an arbitrary discrete state space. For fixed finite range of the Markov chain, we derive an explicit formula for the joint density of all local times on the range, at any fixed time. We use standard tools from the theory of stochastic processes and finite-dimensional complex calculus. We apply this formula in the following directions: (1) we derive large deviation upper estimates for the normalized local times beyond the exponential scale, (2) we derive the upper bound in Varadhan's \chwk{l}emma for any measurable functional of the local times, \ch{and} (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of $\Z^d$ tending to $\Z^d$ as time diverges. We finally discuss the relation of our density formula to the Ray-Knight theorem for continuous-time simple random walk on $\Z$, which is analogous to the well-known Ray-Knight description of Brownian local times. In this extended version, we prove that the Ray-Knight theorem follows from our density formula.