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The existence of a steady state for perturbed symmetric random walk on a random lattice

Komorowski T., Krupa G.

Probability and Mathematical Statistics

2004

Vol. 24, Fasc. 1

121--144

In the present paper we consider a continuous time random walk on an anisotropic random lattice. We show the existence of a steady state ̄μα for the environment process (ζ(t))t ≥ 0 corresponding to the walk. This steady state has the property that the ergodic averages of (F(ζ(t)))t ≥ 0, where F is local (i.e. it depends on finitely many bonds of the lattice only), converge almost surely in the annealed measure to ∫Fd ̄μα.