Scaling random walks on arbitrary sets

Let I be a countably infinite set of points in [open face R] which we can write as I={ui: i[set membership][open face Z]}, with ui<ui+1 for every i and where ui[rightward arrow]±[infty infinity] if i[rightward arrow]±[infty infinity]. Consider a continuous-time Markov chain Y={Y(t): t[gt-or-equal, slanted]0} with state space I such that:

Y is driftless; and

Y jumps only between nearest neighbours.

We remember that the simple symmetric random-walk, when repeatedly rescaled suitably in space and time, looks more and more like a Brownian motion. In this paper we explore the convergence properties of the Markov chain Y on the set I under suitable space-time scalings. Later, we consider some cases when the set I consists of the points of a renewal process and the jump rates assigned to each state in I are perhaps also randomly chosen.

This work sprang from a question asked by one of us (Sibson) about ‘driftless nearest-neighbour’ Markov chains on countable subsets I of [open face R]d, work of Sibson [7] and of Christ, Friedberg and Lee [2] having identified examples of such chains in terms of the Dirichlet tessellation associated with I. Amongst methods which can be brought to bear on this d-dimensional problem is the theory of Dirichlet forms. There are potential problems in doing this because we wish I to be random (for example, a realization of a Poisson point process), we do not wish to impose artificial boundedness conditions which would clearly make things work for certain deterministic sets I. In the 1-dimensional case discussed here and in the following paper by Harris, much simpler techniques (where we embed the Markov chain in a Brownian motion using local time) work very effectively; and it is these, rather than the theory of Dirichlet forms, that we use.