Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2022
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866913430934388736 |
|---|---|
| author | Knorst, J. Lopes, A. O. Muller, G. Neumann, A. |
| author_facet | Knorst, J. Lopes, A. O. Muller, G. Neumann, A. |
| contents | Consider the semi-flow given by the continuous time shift $Θ_t:\mathcal{D} \to \mathcal{D} $, $t \geq 0$, acting on the
$\mathcal{D} $ of \textit{càdlàg} paths $w: [0,\infty) \to S^1$, where $S^1$ is the unitary circle. We equip the space $\mathcal{D} $ with the Skorokhod metric, and we show that the semi-flow is expanding. We also introduce a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ acts linearly on continuous functions $f:S^1\to\mathbb{R}$. This stochastic semigroup and an initial vector of probability $π$ define an associated stationary shift-invariant probability $\mathbb{P}$ on the Polish space $\mathcal{D} $. Given such $\mathbb{P}$ and an Hölder potential $V:S^1 \to \mathbb{R}$, we define a continuous time Ruelle operator, which is described by a family of linear operators $ \mathbb{L}^t_V$, $t\geq 0,$ acting on continuous functions $φ: S^1 \to \mathbb{R}$. More precisely, given any Hölder $V$ and $t\geq 0$, the operator $ \mathbb{L}^t_V$, is defined by
$φ\to ψ(y) = \mathbb{L}^t_V(φ)(y)= \int_{w(t)=y} e^{ \int_0^t V(w(s)) ds} φ(w(0)) d \mathbb{P}(w).$
For some specific parameters we show the existence of an eigenvalue $λ_V$ and an associated Hölder eigenfunction $φ_V>0$.After a coboundary procedure we obtain another stochastic semigroup, with infinitesimal generator $L_V$, and this will define a new probability $\mathbb{P}_V$ on $\mathcal{D}$, which we call the Gibbs (or, equilibrium) probability for the potential $V$. In this case, we define entropy for some shift-invariant probabilities on $\mathcal{D}$, and we consider a variational problem of pressure. Finally, we define entropy production and present our main result: we analyze its relation with time-reversal and symmetry of $L$. We also show that the continuous-time shift $Θ_t$, acting on the Skorohod space $D$, is expanding. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_01989 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness Knorst, J. Lopes, A. O. Muller, G. Neumann, A. Dynamical Systems Statistical Mechanics Mathematical Physics Probability 37A50, 37A10, 37A60, 37D35, 60G10, 60J25, 82B05, 82B30 Consider the semi-flow given by the continuous time shift $Θ_t:\mathcal{D} \to \mathcal{D} $, $t \geq 0$, acting on the $\mathcal{D} $ of \textit{càdlàg} paths $w: [0,\infty) \to S^1$, where $S^1$ is the unitary circle. We equip the space $\mathcal{D} $ with the Skorokhod metric, and we show that the semi-flow is expanding. We also introduce a stochastic semi-group $e^{t\, L}$, $t \geq 0,$ where $L$ acts linearly on continuous functions $f:S^1\to\mathbb{R}$. This stochastic semigroup and an initial vector of probability $π$ define an associated stationary shift-invariant probability $\mathbb{P}$ on the Polish space $\mathcal{D} $. Given such $\mathbb{P}$ and an Hölder potential $V:S^1 \to \mathbb{R}$, we define a continuous time Ruelle operator, which is described by a family of linear operators $ \mathbb{L}^t_V$, $t\geq 0,$ acting on continuous functions $φ: S^1 \to \mathbb{R}$. More precisely, given any Hölder $V$ and $t\geq 0$, the operator $ \mathbb{L}^t_V$, is defined by $φ\to ψ(y) = \mathbb{L}^t_V(φ)(y)= \int_{w(t)=y} e^{ \int_0^t V(w(s)) ds} φ(w(0)) d \mathbb{P}(w).$ For some specific parameters we show the existence of an eigenvalue $λ_V$ and an associated Hölder eigenfunction $φ_V>0$.After a coboundary procedure we obtain another stochastic semigroup, with infinitesimal generator $L_V$, and this will define a new probability $\mathbb{P}_V$ on $\mathcal{D}$, which we call the Gibbs (or, equilibrium) probability for the potential $V$. In this case, we define entropy for some shift-invariant probabilities on $\mathcal{D}$, and we consider a variational problem of pressure. Finally, we define entropy production and present our main result: we analyze its relation with time-reversal and symmetry of $L$. We also show that the continuous-time shift $Θ_t$, acting on the Skorohod space $D$, is expanding. |
| title | Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness |
| topic | Dynamical Systems Statistical Mechanics Mathematical Physics Probability 37A50, 37A10, 37A60, 37D35, 60G10, 60J25, 82B05, 82B30 |
| url | https://arxiv.org/abs/2208.01989 |