Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator
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| Formato: | Preprint |
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2022
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| author | Lopes, A. O. Muller, G. Neumann, A. |
| author_facet | Lopes, A. O. Muller, G. Neumann, A. |
| contents | We consider a Riemmaniann compact manifold $M$, the associated Laplacian $Δ$ and the corresponding Brownian motion $X_t$, $t\geq 0.$ Given a Lipschitz function $V:M\to\mathbb R$ we consider the operator $\frac{1}{2}Δ+V$, which acts on differentiable functions $f: M\to\mathbb R$ via the operator
$$\frac{1}{2} Δf(x)+\,V(x)f(x) ,$$ 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 $λ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_01993 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator Lopes, A. O. Muller, G. Neumann, A. Probability Statistical Mechanics Mathematical Physics Dynamical Systems 60J25, 60J60, 60J65, 58J65, 37D35 We consider a Riemmaniann compact manifold $M$, the associated Laplacian $Δ$ and the corresponding Brownian motion $X_t$, $t\geq 0.$ Given a Lipschitz function $V:M\to\mathbb R$ we consider the operator $\frac{1}{2}Δ+V$, which acts on differentiable functions $f: M\to\mathbb R$ via the operator $$\frac{1}{2} Δf(x)+\,V(x)f(x) ,$$ 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 $λ$ 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. |
| title | Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator |
| topic | Probability Statistical Mechanics Mathematical Physics Dynamical Systems 60J25, 60J60, 60J65, 58J65, 37D35 |
| url | https://arxiv.org/abs/2208.01993 |