Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness

Abstract

Consider the semi-flow given by the continuous time shift $\Theta_t:\mathcal{D} \to \mathcal{D}$, $t \geq 0$, acting on the Skorokhod space $\mathcal{D}$, of paths $w: [0,\infty) \to S^1$, where $S^1$ is the unitary circle We show that the semi-flow is expanding. We consider a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ (the infinitesimal generator) This stochastic semi-group and an initial vector of probability $\pi$ defines an associated stationary shift-invariant probability $\mathbb{P}$. This probability $\mathbb{P}$ will play the role of an {\it a priori} probability. Given such $\mathbb{P}$ and a H\"older potential $V:S^1 \to \mathbb{R}$, we define a continuous time Ruelle operator: a family of linear operators $\mathbb{L}^t_V$, $t\geq 0,$ acting on continuous functions $\varphi: S^1 \to \mathbb{R}$, defined by $\varphi \,\to \psi(y) = \mathbb{L}^t_V(\varphi)(y)= \int_{w(t)=y} e^{ \int_0^t V(w(s)) \, ds} \, \varphi (w(0)) \,d \mathbb{P}(w).\,$ We show the existence of an eigenvalue $\lambda_V$ and an associated H\"older eigenfunction $\varphi_V>0$ for the semi-group $\mathbb{L}_V^t$, $t\geq 0.$ After a coboundary procedure we obtain another stochastic semi-group, with infinitesimal generator $L_V$, and this will define a new probability $\mathbb{P}_V$ on $\mathcal{D}$, which we call the Gibbs (or, equilibrium) probability for the potential $V$. We define entropy, for some shift-invariant probabilities on $\mathcal{D}$, and we consider a variational problem of pressure. Finally, we define entropy production and we analyze its relation with time-reversal and symmetry of $L$. We wonder if the point of view described here provides a sketch (as an alternative for the Anosov one) for the chaotic hypothesis for a particle system held in a nonequilibrium stationary state, as delineated by Ruelle, Gallavotti, and Cohen