Walsh's Brownian Motion and Donsker Scaling Limits of Perturbed Random Walks

2023/10/16 by Ilya Pavlyukevich, Pavlyukevich, Ilya, Andrey Pilipenko +1 · 1 citation
Mathematics · #60F17 #60G42 #60G50 #60H10 #60J10 #60J55 #60K37 #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Mathematical Dynamics and Fractals #Probability (math.PR) #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2310.10809

openalex publication_date 2023/10/16 · openalex created_date 2023/10/20 · openalex updated_date 2026/07/28

Abstract

In this paper we study Markov chains with the state space given by the coordinate axes of \mathbb R^m, m ≥ 2, whose step sizes on each positive half-axis are distributed according to a centered probability distribution with variance v_i² ∈ (0, ∞), i = 1,…, m. Under very mild assumptions on the jumps sizes on the negative half-axes, we show that the Donsker scaling limit of such Markov chains is a Walsh Brownian motion whose weights are determined explicitly in terms of stationary distributions of certain embedded Markov chains. This convergence result is applied to integer-valued random walks perturbed on a finite subset of \mathbb Z called a membrane. We show that their Donsker scaling limit is an oscillating skew Brownian motion.

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