关于解耦马尔可夫群猜想
On the decoupled Markov group conjecture
摘要：
马尔可夫群猜想是可数状态空间马尔可夫过程理论中的一个长期悬而未决的问题，它证明了在$\ell^1$上的强连续马尔可夫半群$T = (T_t)_{t \in [0,\infty)}$有界生成元，如果算子$T_1$是双射的.反驳该猜想的尝试通常旨在将维数增长的有限维矩阵半群粘合在一起-即，它试图表明，马尔可夫群猜想是错误的，即使是马尔可夫过程，解耦成（无限多）有限维系统。在这篇文章中，我们表明，这种尝试必然会失败，即，我们证明了马尔可夫群猜想的过程，在上述方式解耦。事实上，我们甚至给出了一个更一般的结果，给出了一个普遍的范数估计的有界生成元$Q$的任何巴拿赫格上的正半群。我们的证明是基于过滤器的产品技术，无限维Perron-Frobenius理论和Gelfand的$T = \operatorname{id}$定理。
英文摘要：
The Markov group conjecture, a long-standing open problem in the theory of Markov processes with countable state space, asserts that a strongly continuous Markov semigroup $T = (T_t)_{t \in [0,\infty)}$ on $\ell^1$ has bounded generator if the operator $T_1$ is bijective. Attempts to disprove the conjecture have often aimed at glueing together finite dimensional matrix semigroups of growing dimension - i.e., it was tried to show that the Markov group conjecture is false even for Markov processes that decouple into (infinitely many) finite dimensional systems. In this article we show that such attempts must necessarily fail, i.e., we prove the Markov group conjecture for processes that decouple in the way described above. In fact, we even show a more general result that gives a universal norm estimate for bounded generators $Q$ of positive semigroups on any Banach lattice. Our proof is based on a filter product technique, infinite dimensional Perron-Frobenius theory and Gelfand's $T = \operatorname{id}$ theorem.
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