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Functional limit theorems for perturbed random walks
Abstract
We consider scaling limits of integer-valued random walks whose transition probabilities are translation-invariant everywhere except on a set that we will call a membrane or an interface. If the jumps outside the membrane are i.i.d. mean-zero random variables with finite variance, then possible examples of the corresponding limit processes include reflected Brownian motion, skew Brownian motion, and Brownian motion with a jump-type exit from zero. We also discuss the question of what the possible limits are if the jumps outside the membrane belong to the domain of attraction of a stable law.
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BibTeX
@misc{pilipenko2026functional,
title = {{Functional limit theorems for perturbed random walks}},
author = {Andrey Pilipenko},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17262},
eprint = {2609.17262},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17262},
}
RIS
TY - UNPB
TI - Functional limit theorems for perturbed random walks
AU - Andrey Pilipenko
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17262
UR - https://arxiv.org/abs/2609.17262
AB - We consider scaling limits of integer-valued random walks whose transition probabilities are translation-invariant everywhere except on a set that we will call a membrane or an interface. If the jumps outside the membrane are i.i.d. mean-zero random variables with finite variance, then possible examples of the corresponding limit processes include reflected Brownian motion, skew Brownian motion, and Brownian motion with a jump-type exit from zero. We also discuss the question of what the possible limits are if the jumps outside the membrane belong to the domain of attraction of a stable law.
ER -