On a limit behavior of a sequence of Markov processes perturbed in a neighborhood of a singular point

TL;DR

This paper investigates the limiting behavior of Markov processes near a singular point, demonstrating that under certain perturbations, the processes converge to a skew Brownian motion, revealing new insights into their asymptotic properties.

Contribution

The paper establishes a general invariance principle for Markov processes perturbed near a singular point, with specific analysis of symmetric random walks leading to skew Brownian motion.

Findings

Processes outside the neighborhood converge to a probability law.

Perturbed symmetric random walk converges to skew Brownian motion.

Standard scaling yields the invariance principle.

Abstract

We study a limit behavior of a sequence of Markov processes (or Markov chains) such that their distributions outside of any neighborhood of a "singular" point attract to some probability law. In any neighborhood of this point the behavior may be irregular. As an example of the general result we consider a symmetric random walk with the unit jump that is perturbed in a neighborhood of 0. The invariance principle is obtained under standard scaling of time and space. The limit process turns out to be a skew Brownian motion.