具有双面膜的Ζm上随机漫步的极限行为

IF 0.6 4区 数学 Q4 STATISTICS & PROBABILITY Esaim-Probability and Statistics Pub Date : 2022-07-31 DOI:10.1051/ps/2022009
V. Bogdanskii, I. Pavlyukevich, A. Pilipenko
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引用次数: 2

摘要

我们研究了在超平面(膜)H = {0} × zm -1外表现为标准对称随机游走的zm, m≥2上的马尔可夫链。膜的出口概率(穿透概率)是周期性的,也取决于进入的方向,这使得膜是双面的。此外,允许沿膜滑动。我们证明了这种马尔可夫链的自然尺度极限是一个m维扩散,其第一个坐标是一个偏布朗运动,另一个m-1坐标是一个布朗运动,其奇异漂移由第一个坐标的局部时间在0处控制。在证明中,我们利用沃尔什布朗运动的鞅特征,确定了有效渗透率和滑动方向。最后,对于无滑动和随机穿透概率的单侧膜,建立了类似的收敛定理。
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Limit behaviour of random walks on Ζm with two-sided membrane
We study Markov chains on Z m , m ≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane) H = {0} × Z m-1 . The exit probabilities from the membrane (penetration probabilities) H are periodic and also depend on the incoming direction to H, what makes the membrane H two-sided. Moreover, sliding along the membrane is allowed. We show that the natural scaling limit of such Markov chains is a m-dimensional diffusion whose first coordinate is a skew Brownian motion and the other m-1 coordinates is a Brownian motion with a singular drift controlled by the local time of the first coordinate at 0. In the proof we utilize a martingale characterization of the Walsh Brownian motion and determine the effective permeability and slide direction. Eventually, a similar convergence theorem is established for the one-sided membrane without slides and random iid penetration probabilities.
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来源期刊
Esaim-Probability and Statistics
Esaim-Probability and Statistics STATISTICS & PROBABILITY-
CiteScore
1.00
自引率
0.00%
发文量
14
审稿时长
>12 weeks
期刊介绍: The journal publishes original research and survey papers in the area of Probability and Statistics. It covers theoretical and practical aspects, in any field of these domains. Of particular interest are methodological developments with application in other scientific areas, for example Biology and Genetics, Information Theory, Finance, Bioinformatics, Random structures and Random graphs, Econometrics, Physics. Long papers are very welcome. Indeed, we intend to develop the journal in the direction of applications and to open it to various fields where random mathematical modelling is important. In particular we will call (survey) papers in these areas, in order to make the random community aware of important problems of both theoretical and practical interest. We all know that many recent fascinating developments in Probability and Statistics are coming from "the outside" and we think that ESAIM: P&S should be a good entry point for such exchanges. Of course this does not mean that the journal will be only devoted to practical aspects.
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