Asymptotic behaviour of a random walk killed on a finite set

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.

16 pages, to appear in Potential Analysis, 2016