Let I be a countably infinite set of points in ℝ
which we can write
as I={ui: i∈ℤ},
with ui<ui+1
for
every i and where ui→±∞
if
i→±∞. Consider
a continuous-time Markov chain Y={Y(t): t[ges ]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
ℝd, 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.