遍历Optimization中的对偶势、对合核和输运
The Dual Potential, the involution kernel and Transport in Ergodic Optimization
摘要：
考虑作用在伯努利空间$Σ={1,2,...,n}^\mathbb{N}$上的移位$σ$。我们用$\hatΣ= {1,2,...,n}^\mathbb{Z}$表示。我们分析了Holder势$A: Σ\to \mathbb{R}$的最大概率$μ_{\infty,A}$的几个性质。通过对合核$W: \hatΣ \to \mathbb{R}$，与$A(x)$相关联，已知可以得到对偶势$A^*(y)$，其中$(x,y)\in \hatΣ$。将$μ_{\infty, A^*}$视为$A^*$的最大化概率。我们想考虑从$μ_{\infty,A}$到$μ_{\infty,A^*}$的传输问题。在这种情况下，考虑成本函数$c(x,y) = I(x) - W(x,y) +γ$是很自然的，其中$I$是偏差函数。Kantorovich运输对偶问题的一对函数是$(-V,-V^*$），其中我们分别用$V$和$V^*$表示两个校准的子动作，对于$A$，用$A^*$表示$μ_{\infty,A}$。分析了最优规划$\hatμ$的图的性质。
英文摘要：
Consider the shift $σ$ acting on the Bernoulli space $Σ={1,2,...,n}^\mathbb{N}$. We denote $\hatΣ= {1,2,...,n}^\mathbb{Z}$. We analyze several properties of the maximizing probability $μ_{\infty,A}$ of a Holder potential $A: Σ\to \mathbb{R}$. Associated to $A(x)$, via the involution kernel, $W: \hatΣ \to \mathbb{R}$, it is known that can we get the dual potential $A^*(y)$, where $(x,y)\in \hatΣ$. Consider $μ_{\infty, A^*}$ a maximizing probability for $A^*$. We would like to consider the transport problem from $μ_{\infty,A}$ to $μ_{\infty,A^*}$. In this case, it is natural to consider the cost function $c(x,y) = I(x) - W(x,y) +γ$, where $I$ is the deviation function. The pair of functions for the Kantorovich Transport dual Problem are $(-V,-V^*$), where we denote the two calibrated sub-actions by $V$ and $V^*$, respectively, for $A$ and $A^*$ for $μ_{\infty,A}$. We analyze the graph property for the optimal plan $\hatμ$.
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