Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator

TL;DR

This paper explores the continuous-time analogue of the Ruelle operator on a Riemannian manifold, linking entropy, Gibbs states, and thermodynamic formalism through spectral analysis of a Schrödinger-type operator.

Contribution

It introduces a continuous-time Ruelle operator framework on manifolds, connecting spectral properties with Gibbs states and establishing a variational principle for pressure.

Findings

The semigroup $P_t^V$ acts as a continuous-time Ruelle operator.

Existence of a positive eigenfunction $F$ and eigenvalue $or the semigroup.

Construction of stationary Gibbs states via spectral data and coboundary transformations.

Abstract

We consider a Riemmaniann compact manifold $M$, the associated Laplacian $\Delta$ and the corresponding Brownian motion $X_t$, $t\geq 0.$ Given a Lipschitz function $V:M\to\mathbb R$ we consider the operator $\frac{1}{2}\Delta+V$, which acts on differentiable functions $f: M\to\mathbb R$ via the operator $$\frac{1}{2} \Delta f(x)+\,V(x)f(x) ,$$ for all $x\in M$. Denote by $P_t^V$, $t \geq 0,$ the semigroup acting on functions $f: M\to\mathbb R$ given by $$P_{t}^V (f)(x)\,:=\, \mathbb E_{x} \big[e^{\int_0^{t} V(X_r)\,dr} f(X_t)\big].\,$$ We will show that this semigroup is a continuous-time version of the discrete-time Ruelle operator. Consider the positive differentiable eigenfunction $F: M \to \mathbb{R}$ associated to the main eigenvalue $\lambda$ for the semigroup $P_t^V$, $t \geq 0$. From the function $F$, in a procedure similar to the one used in the case of discrete-time Thermodynamic Formalism, we can associate via a coboundary procedure a certain stationary Markov semigroup. The probability on the Skhorohod space obtained from this new stationary Markov semigroup can be seen as a stationary Gibbs state associated with the potential $V$. We define entropy, pressure, the continuous-time Ruelle operator and we present a variational principle of pressure for such a setting.