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

Consider the semi-flow given by the continuous time shift \(\Theta _t:{\mathcal {D}} \rightarrow {\mathcal {D}} \) ${\mathrm{\Theta }}_t:\mathcal{D}\to \mathcal{D}$ , \(t \ge 0\) $t\ge 0$ , acting on the \({\mathcal {D}} \) $\mathcal{D}$ of càdlàg paths (right continuous with left limits) \(w: [0,\infty ) \rightarrow S^1\) $w:[0,\infty )\to S^1$ , where \(S^1\) $S^1$ is the unitary circle (one can also take [0, 1] instead of \(S^1\) $S^1$ ). We equip the space \({\mathcal {D}} \) $\mathcal{D}$ with the Skorokhod metric, and we show that the semi-flow is expanding. We also introduce a stochastic semi-group \(e^{t\, L}\) $e^{tL}$ , \(t \ge 0,\) $t\ge 0,$ where L (the infinitesimal generator) acts linearly on continuous functions \(f:S^1\rightarrow {\mathbb {R}}\) $f:S^1\to \mathbb{R}$ . This stochastic semigroup and an initial vector of probability \(\pi \) $\pi$ define an associated stationary shift-invariant probability \({\mathbb {P}}\) $\mathbb{P}$ on the Polish space \({\mathcal {D}} \) $\mathcal{D}$ . This probability \({\mathbb {P}}\) $\mathbb{P}$ will play the role of an a priori probability. Given such \({\mathbb {P}}\) $\mathbb{P}$ and an Hölder potential \(V:S^1 \rightarrow {\mathbb {R}}\) $V:S^1\to \mathbb{R}$ , we define a continuous time Ruelle operator, which is described by a family of linear operators \( {\mathbb {L}}^t_V\) ${\mathbb{L}}_V^t$ , \(t\ge 0,\) $t\ge 0,$ acting on continuous functions \(\varphi : S^1 \rightarrow {\mathbb {R}}\) $\phi :S^1\to \mathbb{R}$ . More precisely, given any Hölder V and \(t\ge 0\) $t\ge 0$ , the operator \( {\mathbb {L}}^t_V\) ${\mathbb{L}}_V^t$ , is defined by \(\begin{aligned} \varphi \,\rightarrow \psi (y) = {\mathbb {L}}^t_V(\varphi )(y)= \int _{w(t)=y} e^{ \int _0^t V(w(s)) \, ds} \, \varphi (w(0)) \,d {\mathbb {P}}(w). \end{aligned}\) $$\begin{array}{c}\hfill \phi \to \psi (y)={\mathbb{L}}_V^t(\phi )(y)={\int }_{w(t)=y}{e}^{{\int }_0^{t}V(w(s))ds}\phi (w(0))d\mathbb{P}(w).\end{array}$$ For some specific parameters we show the existence of an eigenvalue \(\lambda _V\) ${\lambda }_V$ and an associated Hölder eigenfunction \(\varphi _V>0\) ${\phi }_V>0$ for the semigroup \({\mathbb {L}}_V^t\) ${\mathbb{L}}_V^t$ , \(t\ge 0.\) $t\ge 0.$ After a coboundary procedure we obtain another stochastic semigroup, with infinitesimal generator \(L_V\) $L_V$ , and this will define a new probability \({\mathbb {P}}_V\) ${\mathbb{P}}_V$ on \({\mathcal {D}}\) $\mathcal{D}$ , which we call the Gibbs (or, equilibrium) probability for the potential V. In this case, we define entropy for some continuous time shift-invariant probabilities on \({\mathcal {D}}\) $\mathcal{D}$ , 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 \(\Theta _t\) ${\mathrm{\Theta }}_t$ , acting on the Skorokhod space D, is expanding. We wonder if the point of view described here provides a sketch (as an alternative to the Anosov one) for the chaotic hypothesis for a particle system held in a nonequilibrium stationary state, as delineated by Ruelle, Gallavotti, and Cohen.