DIFFUSION PROCESSES: ENTROPY, GIBBS STATES AND THE CONTINUOUS TIME RUELLE OPERATOR

We consider a Riemannian compact manifold M, the associated Laplacian Delta and the corresponding Brownian motion X-t, t >= 0. Given a Lipschitz function V : M -> R we consider the operator 1/2 Delta + V, which acts on differentiable functions f : M -> R via the expression 1/2 Delta f(x) + V (x)f(x), for all x is an element of M. Denote by P-t(V), t >= 0, the semigroup acting on functions f : M -> R given by P-t(V)(f)(x) := E-x [e(integral t0V(Xr) dr)f(X-t)]. We will derive results that show that this semigroup is a continuous-time version of the discrete-time Ruelle operator. Consider the positive differentiable eigenfunction F : M -> R associated with the main eigenvalue lambda, for the semigroup P-t(V), t >= 0. From the function F, in a procedure similar to the one used in discrete-time Thermodynamic Formalism, we can associate by way of a coboundary procedure, a certain stationary Markov semigroup. We show that the probability on the Skorohod space obtained from this new stationary Markov semigroup meets the requirements to be called stationary Gibbs state associated with the potential V. We define entropy, pressure, and the continuous-time Ruelle operator. Also, we present a variational principle of pressure for such a setting.