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Joint density for the local times of continuous-time Markov chains. (English) Zbl 1127.60076

A continuous time Markov chain \((X_t)_{t \in [0,\infty)}\) on the countably infinite or finite set \(\Lambda\) is considered. The main object of the study are local times, defined by \(\ell_T(x)= \int_0^T I_{\{X_s=x\}}\,ds\), where \(I_{\{X_s=x\}}\) is the indicator of the probability event \({\{X_s=x\}}\). For a fixed finite subset \(R \subseteq \Lambda\), an explicit formula is derived for the joint density of all local times \((\ell_T(x))_{x\in\mathbb R}\) at any fixed time \(T\). Authors use standard tools from the theory of stochastic processes and finite-dimensional complex calculus.
Although the formula is rather involved, it luckily provides the easier method to obtain the upper bounds on the density. The authors derive three such bounds: (1) the large deviation upper estimates for the normalized local times \((\frac{1}{T}\ell_T(x))_{x \in \Lambda}\) beyond the exponential scale, (2) the upper bound in Varadhan’s lemma for any measurable functional of the local times and (3) the large deviation upper bounds for continuous-time simple random walks on large subboxes of \(Z^d\) tending to \(Z^d\) as time diverges.
Finally, the relation of the formula to the Ray-Knight theorem for continuous-time simple random walk on \(Z\), which is analogous to the Ray-Knight description of Brownian local times, is discussed.

MSC:

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