## The Annals of Probability

### Joint density for the local times of continuous-time Markov chains

#### 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 lemma for any measurable functional of the local times, and (3) we derive large deviation upper bounds for continuous-time simple random walk on large subboxes of ℤd tending to ℤd 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.

#### Article information

Source
Ann. Probab., Volume 35, Number 4 (2007), 1307-1332.

Dates
First available in Project Euclid: 8 June 2007

https://projecteuclid.org/euclid.aop/1181334246

Digital Object Identifier
doi:10.1214/009171906000001024

Mathematical Reviews number (MathSciNet)
MR2330973

Zentralblatt MATH identifier
1127.60076

#### Citation

Brydges, David; van der Hofstad, Remco; König, Wolfgang. Joint density for the local times of continuous-time Markov chains. Ann. Probab. 35 (2007), no. 4, 1307--1332. doi:10.1214/009171906000001024. https://projecteuclid.org/euclid.aop/1181334246

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