We study Markov chains on ℤ$$, m ≥ 2, that behave like a standard symmetric random walk outside of the hyperplane (membrane) H = {0} × ℤ$$. 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.
Keywords: Skew Brownian motion, Walsh’s Brownian motion, perturbed random walk, two-sided membrane, weak convergence, martingale characterization,
@article{PS_2022__26_1_352_0,
author = {Bogdanskii, Victor and Pavlyukevich, Ilya and Pilipenko, Andrey},
title = {Limit behaviour of random walks on $\mathbb{Z}^m$ with two-sided membrane},
journal = {ESAIM: Probability and Statistics},
pages = {352--377},
year = {2022},
publisher = {EDP-Sciences},
volume = {26},
doi = {10.1051/ps/2022009},
mrnumber = {4481124},
zbl = {1520.60036},
language = {en},
url = {https://www.numdam.org/articles/10.1051/ps/2022009/}
}
TY - JOUR
AU - Bogdanskii, Victor
AU - Pavlyukevich, Ilya
AU - Pilipenko, Andrey
TI - Limit behaviour of random walks on $\mathbb{Z}^m$ with two-sided membrane
JO - ESAIM: Probability and Statistics
PY - 2022
SP - 352
EP - 377
VL - 26
PB - EDP-Sciences
UR - https://www.numdam.org/articles/10.1051/ps/2022009/
DO - 10.1051/ps/2022009
LA - en
ID - PS_2022__26_1_352_0
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%A Pavlyukevich, Ilya
%A Pilipenko, Andrey
%T Limit behaviour of random walks on $\mathbb{Z}^m$ with two-sided membrane
%J ESAIM: Probability and Statistics
%D 2022
%P 352-377
%V 26
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/ps/2022009/
%R 10.1051/ps/2022009
%G en
%F PS_2022__26_1_352_0
Bogdanskii, Victor; Pavlyukevich, Ilya; Pilipenko, Andrey. Limit behaviour of random walks on $\mathbb{Z}^m$ with two-sided membrane. ESAIM: Probability and Statistics, Tome 26 (2022), pp. 352-377. doi: 10.1051/ps/2022009
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