  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.
