The Annals of Probability

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

David Brydges, Remco van der Hofstad, and Wolfgang König

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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

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

First available in Project Euclid: 8 June 2007

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Zentralblatt MATH identifier

Primary: 60J55: Local time and additive functionals 60J27: Continuous-time Markov processes on discrete state spaces

Local times density large deviations upper bound Ray–Knight theorem Varadhan’s lemma


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.

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