论文标题
扩散过程：熵，吉布斯州和连续的时间ruelle操作员
Diffusion Processes: entropy, Gibbs states and the continuous time Ruelle operator
论文作者
论文摘要
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 $f: M\to\mathbb R$通过操作员 $$ \ frac {1} {2}Δf（x）+\，v（x）f（x），$$ in M $中的所有$ x \。 用$ p_t^v $，$ t \ geq 0，$代表函数上的半群$ f：m \ to \ mathbb r $由$ p _ {t}^v（f）（x）\，：= \，：= \，： f（x_t）\ big]。\，$$ 我们将证明该半群是离散时间Ruelle操作员的连续时间版本。 考虑阳性可分解的特征函数$ f：m \ to \ mathbb {r} $与semigroup $ p_t^v $，$ t \ geq 0 $相关的主要eigenvalue $λ$。从功能$ f $的过程中，在类似于离散时间热力学形式主义的过程中，我们可以通过串联程序与某个固定的马尔可夫半群相关联。从这个新的固定马尔可夫半群中获得的Skhorohod空间的概率可以看作是与潜在$ v $相关的固定吉布斯州。我们定义熵，压力，连续的ruelle操作员，并为这种设置提供了压力的变化原理。
We consider a Riemmaniann compact manifold $M$, the associated Laplacian $Δ$ and the corresponding Brownian motion $X_t$, $t\geq 0.$ Given a Lipschitz function $V:M\to\mathbb R$ we consider the operator $\frac{1}{2}Δ+V$, which acts on differentiable functions $f: M\to\mathbb R$ via the operator $$\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.
