We consider a Riemmaniann compact manifold M, the associated Laplacian
Ξ and the corresponding Brownian motion Xtβ, tβ₯0. Given a
Lipschitz function V:MβR we consider the operator
21βΞ+V, which acts on differentiable functions f:MβR via the operator
21βΞf(x)+V(x)f(x), for all xβM.
Denote by PtVβ, tβ₯0, the semigroup acting on functions f:MβR given by PtVβ(f)(x):=Exβ[eβ«0tβV(Xrβ)drf(Xtβ)].
We will show that this semigroup is a continuous-time version of the
discrete-time Ruelle operator.
Consider the positive differentiable eigenfunction F:MβR
associated to the main eigenvalue Ξ» for the semigroup PtVβ, tβ₯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