Space–time continuous limit of random walks with hyperbolic scaling

Abstract

We consider space–time inhomogeneous one-dimensional random walks which move by ±Δx in each time interval Δt with arbitrary transition probabilities depending on position and time. Unlike Donsker’s theorem, we study the continuous limit of the random walks as Δx,Δt→0 under hyperbolic scaling λ1≥Δt/Δx≥λ0>0 with fixed numbers λ1 and λ0. Our aim is to present explicit formulas and estimates of probabilistic quantities which characterize asymptotics of the random walks as Δx,Δt→0. This provides elementary proofs of several limit theorems on the random walks. In particular, if transition probabilities satisfy a Lipschitz condition, the random walks converge to solutions of ODEs. This is the law of large numbers. The results here will be foundations of a stochastic and variational approach to finite difference approximation of nonlinear PDEs of hyperbolic types.