@MISC{Harris_randomwalk, author = {C. Harris and Robin Sibson and David Williams}, title = {Random Walk on an Arbitrary Set}, year = {} }

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Abstract

Let I be a countably infinite set of points in R, and suppose that I has no points of accumulation and that its convex hull is the whole of R. It will be convenient to index I as {ui: i ∈ Z}, with ui < ui+1 for every i. Consider a continuous-time Markov chain Y = {Y (t) : t ≥ 0} on I, with the properties that: Y is driftless; and Y jumps only between nearest neighbours. We propose this as the obvious analogue on an irregular support set of the simple symmetric random walk on the integers; it appears natural to change from a discrete-time to a continuous-time model to accommodate the irregu-larity, so it is not strictly speaking a generalisation. Suitably rescaled in time and space, the simple symmetric random walk converges in law to Brownian motion. In this paper we explore convergence properties for the irregular analogue. In terms of elements of the Q-matrix of Y, the requirement of freedom from drift may be written in a general form as∑ j 6=i qi,j (uj − ui) = 0 for every i. (1.1) Write `i and ri for the gaps to the left and to the right of ui: `i: = ui − ui−1, ri: = ui+1 − ui. Then the limitation to nearest-neighbour jumps, namely qi,j = 0 for j 6 = i − 1, i, i+ 1, 1 allows the immediately off-diagonal elements of Q to be written as qi,i+1 = qi `i `i + ri, qi,i−1 = qi ri `i + ri where qi = −qi,i is the total jump-rate out of state i which we shall not yet fix. Thus even in the regularly spaced case, the change from a discrete-time to a continuous-time model permits some additional generality. Let Ti be the closed interval extending from ui half-way to each of its neighbours ui−1, ui+1, and denote by κi the length of this interval, namely κi: = 12(`i + ri). Define Di as the ‘variance ’ or ‘diffusion ’ coefficient of Y at ui: