Ergodic Theory and Dynamical Systems

Research Article

Approximation by Brownian motion for Gibbs measures and flows under a function

Manfred Denkera1 and Walter Philippa2

a1 Institut für Mathematische Stochastik, Universität Göttingen, Lotzestr. 13 D-3400 Gottingen, West Germany;

a2 Department of Mathematics, University of Illinois, Urbana, IL 61801, USA

Abstract

Let denote a flow built under a Hölder-continuous function l over the base (Σ, μ) where Σ is a topological Markov chain and μ some (ψ-mining) Gibbs measure. For a certain class of functions f with finite 2 + δ-moments it is shown that there exists a Brownian motion B(t) with respect to μ and σ2 > 0 such that μ-a.e.

S0143385700002637_eqnU1

for some 0 < λ < 5δ/588. One can also approximate in the same way by a Brownian motion B*(t) with respect to the probability . From this, the central limit theorem, the weak invariance principle, the law of the iterated logarithm and related probabilistic results follow immediately. In particular, the result of Ratner ([6]) is extended.

(Received August 05 1983)