论文标题
在分离的马尔可夫组猜想上
On the decoupled Markov group conjecture
论文作者
论文摘要
马尔可夫集团的猜想是马尔可夫（Markov）进程理论的一个长期开放问题，他断言，在$ \ ell^1 $上，强烈连续的Markov Semigroup $ t =（t_t）_ {t_t）_ {t \ in [0，\ infty）} $ in $ \ ell^1 $具有界限生成器，如果操作员$ t_1 $ t_1 $是界限。试图反驳猜想的尝试通常旨在将有限维度的有限尺寸矩阵半群粘合在一起 - 即，它试图表明马尔可夫组的猜想是错误的，即使对于将（无限多）有限尺寸系统脱离的马尔可夫过程也是错误的。 在本文中，我们表明，这样的尝试一定要失败，即，我们证明了马尔可夫组以上述方式解散的过程的猜想。实际上，我们甚至展示了更一般的结果，该结果为有限发电机提供了普遍的规范估计，$ q $ q $ q $ q $ q $ q $ q $ q $ q。 我们的证明是基于过滤产品技术，无限的尺寸Perron-frobenius理论和Gelfand的$ t = \ permatatorName {id} $定理。
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.
