论文标题
Skorokhod空间上的热力学形式主义：连续的时间Ruelle操作员，熵，压力，熵产生和扩张性
Thermodynamic Formalism on the Skorokhod space: the continuous time Ruelle operator, entropy, pressure, entropy production and expansiveness
论文作者
论文摘要
考虑连续时间偏移给出的半流$θ_t：\ Mathcal {d} \ to \ Mathcal {d} $，$ t \ geq 0 $，作用于 \ textit {càdlàg}路径$ w：[0，\ infty）\ to s^1 $的$ \ mathcal {d} $，其中$ s^1 $是单一圆圈。我们将Space $ \ Mathcal {D} $配备了Skorokhod度量，并证明半流量正在扩展。我们还引入了一个随机的半组$ e^{t \，l} $，$ t \ geq 0，$其中$ l $在连续函数上线性地作用$ f：s^1 \ to \ mathbb {r} $。这个随机的半群和概率$π$的初始向量定义了波兰空间$ \ mathcal {d} $上关联的固定转换概率$ \ mathbb {p} $。给定这样的$ \ mathbb {p} $和一个hölder的潜在$ v：s^1 \ to \ mathbb {r} $，我们定义了一个连续的时间ruelle运算符，这是由线性运算符$ \ mathbb {l}^t_v $，$ t_v $，$ t \ geq 0，$ t \ geq 0，continume partions $ contectuncomections $ contocy的$ conc. \ mathbb {r} $。更确切地说，给定任何Hölder$ v $和$ t \ geq 0 $，操作员$ \ mathbb {l}^t_v $由 $φ\至ψ（y）= \ mathbb {l}^t_v（φ）（y）（y）= \ int_ {w（t）= y} e^{\ int_0^t v（w（s）ds}φ（w（s））ds}φ（w（0）） 对于某些特定参数，我们显示了特征值$λ_v$和相关的höldereigenFunction $φ_V> 0 $。在共同过程后，我们获得了另一个随机的半群，带有Infititesimal Generator $ l_v $，这将定义一个新的可能性$ \ new概率$ \ v $ nc.我们称之为潜在$ v $的吉布斯（或，平衡）概率。在这种情况下，我们为$ \ Mathcal {d} $上的某些偏移不变的概率定义熵，我们考虑了压力的变化问题。最后，我们定义熵生产并提出我们的主要结果：我们分析了它与$ l $的时间反转和对称性的关系。我们还表明，在Skorohod Space $ D $上作用的连续时间偏移$θ_T$正在扩大。
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.
