预解式的不变测度的不可约常返性、遍历性和极值性
Irreducible recurrence, ergodicity, and extremality of invariant measures for resolvents
摘要：
我们分析了核的一般次马尔可夫预解式的瞬时性、常返性和不可约性，并讨论了它们在固定次不变测度$m$下的性质。我们给出了不变函数的统一刻画，揭示了一个事实：一个$L^p$ -可积函数是调和函数当且仅当它关于弱对偶预解式是调和函数.我们的方法是基于潜在的理论技术弱对偶预解式。我们证明了预解式的$m$ -不可约递归与$m$在所有不变测度集合中的极值之间的等价性，并将这个结果应用于吉布斯态的极值.我们还表明，我们的结果可以适用于非对称狄利克雷形式，在一般情况下，在具体情况下。第二个应用是所谓的福岛遍历定理对称狄利克雷形式的情况下，次马尔可夫预解的内核的扩展。
英文摘要：
We analyze the transience, recurrence, and irreducibility properties of general sub- Markovian resolvents of kernels and their duals, with respect to a fixed sub-invariant measure $m$. We give a unifying characterization of the invariant functions, revealing the fact that an $L^p$-integrable function is harmonic if and only if it is harmonic with respect to the weak dual resolvent. Our approach is based on potential theoretical techniques for resolvents in weak duality. We prove the equivalence between the $m$-irreducible recurrence of the resolvent and the extremality of $m$ in the set of all invariant measures, and we apply this result to the extremality of Gibbs states. We also show that our results can be applied to non-symmetric Dirichlet forms, in general and in concrete situations. A second application is the extension of the so called Fukushima ergodic theorem for symmetric Dirichlet forms to the case of sub-Markovian resolvents of kernels.
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