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Functional limit theorems for perturbed random walks

Andrey Pilipenko

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Source: arXiv

Published: Sep 15, 2026

arXiv: 2609.17262

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Source 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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Functional limit theorems for perturbed random walks — Mathematical Frontier Network