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

TL;DR

This paper develops a thermodynamic formalism for continuous-time semi-flows on Skorokhod space, introducing a Ruelle operator, entropy, pressure, and analyzing entropy production and expansiveness in this setting.

Contribution

It extends thermodynamic formalism to continuous-time semi-flows on Skorokhod space, defining a Ruelle operator, Gibbs measures, and analyzing entropy production and expansiveness.

Findings

Existence of eigenvalues and eigenfunctions for the Ruelle operator.

Definition of Gibbs (equilibrium) probabilities for the potential V.

Analysis of entropy production and its relation to time-reversal symmetry.

Abstract

Consider the semi-flow given by the continuous time shift $\Theta_t:\mathcal{D} \to \mathcal{D} $, $t \geq 0$, acting on the $\mathcal{D} $ of \textit{c\`{a}dl\`{a}g} paths $w: [0,\infty) \to S^1$, where $S^1$ is the unitary circle. We equip the space $\mathcal{D} $ with the Skorokhod metric, and we show that the semi-flow is expanding. We also introduce a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ acts linearly on continuous functions $f:S^1\to\mathbb{R}$. This stochastic semigroup and an initial vector of probability $\pi$ define an associated stationary shift-invariant probability $\mathbb{P}$ on the Polish space $\mathcal{D} $. Given such $\mathbb{P}$ and an H\"older potential $V:S^1 \to \mathbb{R}$, we define a continuous time Ruelle operator, which is described by a family of linear operators $ \mathbb{L}^t_V$, $t\geq 0,$ acting on continuous functions $\varphi: S^1 \to \mathbb{R}$. More precisely, given any H\"older $V$ and $t\geq 0$, the operator $ \mathbb{L}^t_V$, is defined by $\varphi \to \psi(y) = \mathbb{L}^t_V(\varphi)(y)= \int_{w(t)=y} e^{ \int_0^t V(w(s)) ds} \varphi (w(0)) d \mathbb{P}(w).$ For some specific parameters we show the existence of an eigenvalue $\lambda_V$ and an associated H\"older eigenfunction $\varphi_V>0$.After a coboundary procedure we obtain another stochastic semigroup, 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 $V$. In this case, 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 present our main result: we analyze its relation with time-reversal and symmetry of $L$. We also show that the continuous-time shift $\Theta_t$, acting on the Skorohod space $D$, is expanding.