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تسجيل مستخدم جديدA Ruelle Operator for continuous time Markov Chains
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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 $X_t$, $tgeq 0$, 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 ${cal L}^t, tgeq 0$, 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 $tilde{{cal L}}_V^t, tgeq 0$. We are able to show the existence of an eigenfunction $u$ and an eigen-probability $ u_V$ on $Omega$ associated to $tilde{{cal L}}^t_V, tgeq 0$. We also show the following property for the probability $ u_V$: for any integrable $gin {cal L}^infty (P)$ and any real and positive $t$ $$ int e^{-int_0^t (V circ Theta_s)(.) ds} [ (tilde{{cal L}}^t_V (g)) circ theta_t ] d u_V = int g d u_V$$ 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).
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