Local Central Limit Theorem for a Random Walk Perturbed in One Point

TL;DR

This paper proves a local central limit theorem for a symmetric lattice random walk with a one-point antisymmetric perturbation, revealing short-range and long-range corrections to diffusive behavior in different dimensions.

Contribution

It introduces a novel analysis of a perturbed random walk, establishing the local central limit theorem under antisymmetric perturbations.

Findings

Short-range correction to diffusive behavior in all dimensions

Long-range correction specifically in one dimension

Validation of the local central limit theorem for the perturbed process

Abstract

We consider a symmetric random walk on the $\nu$-dimensional lattice, whose exit probability from the origin is modified by an antisymmetric perturbation and prove the local central limit theorem for this process. A short-range correction to diffusive behaviour appears in any dimension along with a long-range correction in the one-dimensional case.