A Ruelle Operator for continuous time Markov Chains

TL;DR

This paper extends the Ruelle theorem to continuous time Markov chains, establishing the existence of eigenfunctions and eigen-probabilities for a modified Ruelle operator, with implications for mathematical physics and $C^*$-algebras.

Contribution

It generalizes the Ruelle theorem to continuous time Markov chains and constructs eigenfunctions and eigen-probabilities for associated operators, relevant for future mathematical physics applications.

Findings

Existence of eigenfunction and eigen-probability for the modified Ruelle operator.

A key integral equation generalizing discrete to continuous time systems.

Relevance to understanding KMS states in $C^*$-algebras.

Abstract

We consider a generalization of the Ruelle theorem for the case of continuous time problems. We present a result which we believe is important for future use in problems in Mathematical Physics related to $C^*$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $t\geq 0$, taking values on S. We denote by $\Omega$ the set of paths $w$ taking values on S (the elements $w$ are locally constant with left and right limits and are also right continuous on $t$). We consider an infinitesimal generator $L$ and a stationary vector $p_0$. We denote by $P$ the associated probability on ($\Omega, {\cal B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${\cal L}^\infty (P)$. From the probability $P$ we define a Ruelle operator ${\cal L}^t, t\geq 0$, acting on functions $f:\Omega \to \mathbb{R}$ of ${\cal L}^\infty (P)$. Given $V:\Omega \to \mathbb{R}$, such that is constant in sets of the form $\{X_0=c\}$, we define a modified Ruelle operator $\tilde{{\cal L}}_V^t, t\geq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $\nu_V$ on $\Omega$ associated to $\tilde{{\cal L}}^t_V, t\geq 0$. We also show the following property for the probability $\nu_V$: for any integrable $g\in {\cal L}^\infty (P)$ and any real and positive $t$ $$ \int e^{-\int_0^t (V \circ \Theta_s)(.) ds} [ (\tilde{{\cal L}}^t_V (g)) \circ \theta_t ] d \nu_V = \int g d \nu_V$$ This equation generalize, for the continuous time Markov Chain, a similar one for discrete time systems (and which is quite important for understanding the KMS states of certain $C^*$-algebras).