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

Abstract

We consider a Riemmaniann compact manifold $M$, the associated Laplacian $\hat{I}”$ and the corresponding Brownian motion $X_t$, $t\ge 0.$ Given a Lipschitz function $V:M\to \mathbb{R}$ we consider the operator $\frac{1}{2}\hat{I}”+V$, which acts on differentiable functions $f:M\to \mathbb{R}$ via the operator $$\frac{1}{2}\hat{I}”f(x)+V(x)f(x),$$ for all $x\in M$. Denote by $P_t^V$, $t\ge 0,$ the semigroup acting on functions $f:M\to \mathbb{R}$ given by $$P_t^V(f)(x):={\mathbb{E}}_x[e^{{\int }_0^{t}V(X_r)dr}f(X_t)].$$ 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 $\hat{I}»$ for the semigroup $P_t^V$, $t\ge 0$. 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.

In section 3 we present a more precise version of what is necessary to use in the previous part of the text