2016/03/07 by Kôhei Uchiyama, Uchiyama, Kohei
Mathematics · Physics and Astronomy · #FOS: Mathematics #Markov Chains and Monte Carlo Methods #Probability (math.PR) #Stochastic processes and statistical mechanics #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1603.02117
openalex publication_date 2016/03/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the transition probability, say pAn(x,y), of a one-dimensional random walk on the integer lattice killed when entering into a non-empty finite set A. The random walk is assumed to be irreducible and have zero mean and a finite variance σ2. We derive the asymptotic form of pAn(x, y) for large n valid uniformly in the regime characterized by the conditions |x|\vee |y| =O(√ n) and |x|\wedge |y|= o(√ n), in which pAt(\bf x,\bf y) behaves for large n like [gA+(x) gA +(y) + gA-(x) gA -(y)] (σ2/2n) pn(y-x). Here pn(y-x) is the transition kernel of the random walk (without killing); g^±A are the Green functions for the "exterior" of A with "pole at ± ∞" normalized so that g^±A(x) ∼ 2|x|/σ2 as x → ±∞; and gA ± are the corresponding Green functions for the time-reversed walk.