具有约束势的离散时间Feynman-Kac算子的谐波函数衰减

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY Alea-Latin American Journal of Probability and Mathematical Statistics Pub Date : 2021-09-08 DOI:10.30757/alea.v19-44
W. Cygan, K. Kaleta, Mateusz 'Sliwi'nski
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引用次数: 1

摘要

我们提出并研究了在可数无限空间中具有限制势的经典Feynman—Kac半群的离散时间对应物。对于一类满足直接阶跃性质的长程马尔可夫链,我们证明了关于离散Feynman—Kac算子的无穷集(次、超)调和函数的尖锐估计。这些结果与在有限几何图形上进化的最近邻随机漫步的情况下的各自估计进行了比较。我们还讨论了在涉及图拉普拉斯算子的方程解的衰减率和离散费曼-卡兹算子的本征函数中的应用。我们包括基于最近邻拉普拉斯算子的分数幂的非局部离散Schr\ odinger算子和相关的准相对论算子等例子。最后,分析了具有直接阶跃性质的各类马尔可夫链,并用实例说明了所得结果。
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Decay of harmonic functions for discrete time Feynman–Kac operators with confining potentials
We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schr\"odinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.
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来源期刊
CiteScore
1.10
自引率
0.00%
发文量
48
期刊介绍: ALEA publishes research articles in probability theory, stochastic processes, mathematical statistics, and their applications. It publishes also review articles of subjects which developed considerably in recent years. All articles submitted go through a rigorous refereeing process by peers and are published immediately after accepted. ALEA is an electronic journal of the Latin-american probability and statistical community which provides open access to all of its content and uses only free programs. Authors are allowed to deposit their published article into their institutional repository, freely and with no embargo, as long as they acknowledge the source of the paper. ALEA is affiliated with the Institute of Mathematical Statistics.
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