A Ruelle Operator for continuous time Markov Chains

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 $C^{\ast }$-Algebras We consider a finite state set $S$ and a stationary continuous time Markov Chain $X_t$, $t\ge 0$, 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 ($\Omega ,\mathcal{B}$). This is the a priori probability. All functions $f$ we consider bellow are in the set ${\mathcal{L}}^{\infty }(P)$. From the probability $P$ we define a Ruelle operator ${\mathcal{L}}^t,t\ge 0$, acting on functions $f:\Omega \to \mathbb{R}$ of ${\mathcal{L}}^{\infty }(P)$. Given $V:\Omega \to \mathbb{R}$, such that is constant in sets of the form $\{X_0=c\}$, we define a modified Ruelle operator ${\tilde{\mathcal{L}}}_V^t,t\ge 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability ${\nu }_V$ on $\Omega$ associated to ${\tilde{\mathcal{L}}}_V^t,t\ge 0$. We also show the following property for the probability ${\nu }_V$: for any integrable $g\in {\mathcal{L}}^{\infty }(P)$ and any real and positive $t$ $$\int e^{-{\int }_0^t(V\circ {\Theta }_s)(.)ds}[({\tilde{\mathcal{L}}}_V^t(g))\circ {\theta }_t]d{\nu }_V=\int gd{\nu }_V$$ 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 $C^{\ast }$-algebras).