Asymptotic behaviour of a random walk killed on a finite set

TL;DR

This paper analyzes the long-term behavior of a two-dimensional random walk that is terminated upon entering a finite set, revealing its asymptotic transition probabilities and harmonic functions.

Contribution

It provides a detailed asymptotic formula for the transition probabilities of the killed random walk, including the harmonic functions involved, extending understanding of such processes.

Findings

Transition probability behaves like a scaled product involving harmonic functions and the original kernel.

Asymptotic behavior is uniform in the parabolic regime where positions are of order √n.

Harmonic functions satisfy specific boundary conditions and grow logarithmically at infinity.

Abstract

We study asymptotic behavior, for large time $n$, of the transition probability of a two-dimensional random walk killed when entering into a non-empty finite subset $A$. We show that it behaves like $4 \tilde u_A(x) \tilde u_{-A}(-y) (\lg n)^{-2} p^n(y- x)$ for large $n$, uniformly in the parabolic regime $|x|\vee |y| =O(\sqrt n)$, where $p^n(y-x)$ is the transition kernel of the random walk (without killing) and $\tilde u_A$ is the unique harmonic function in the 'exterior of $A$' satisfying the boundary condition $\tilde u_A(x) \sim \lg |x|$ at infinity.