[Paper Review] Noncommutative Ergodic Theorems

Anders Karlsson, François Ledrappier|arXiv (Cornell University)|Oct 31, 2011

Geometric and Algebraic Topology11 references20 citations

TL;DR

This paper establishes noncommutative ergodic theorems for products of isometries on proper metric spaces, showing that under integrable displacement, typical trajectories either exhibit sublinear diffusion or converge to a random geodesic ray in the metric compactification. The key contribution is a generalization of the classical Birkhoff and Kingman ergodic theorems to noncommutative settings, with applications to random walks on groups and Brownian motion on Riemannian covers.

ABSTRACT

We present recent results about the asymptotic behavior of ergodic products of isometries of a metric space X. If we assume that the displacement is integrable, then either there is a sublinear diffusion or there is, for almost every trajectory in X, a preferred direction at the boundary. We discuss the precise statement when X is a proper metric space and compare it with classical ergodic theorems. Applications are given to ergodic theorems for nonintegrable functions, random walks on groups and Brownian motion on covering manifolds.

Motivation & Objective

• To generalize classical ergodic theorems to noncommutative settings involving products of isometries on metric spaces.
• To establish conditions under which trajectories of random isometries converge to a preferred direction in the metric compactification.
• To connect the Liouville property of Riemannian covers to the rate of escape of Brownian motion and random walks.
• To extend results from continuous-time diffusion processes (Brownian motion) to discrete-time random walks via discretization.
• To prove that in locally compact groups with proper metrics, the linear drift of a random walk arises entirely from a character if the Liouville property holds.

Proposed method

• Use the metric compactification of a proper metric space to define a notion of 'direction at infinity' for trajectories.
• Apply the subadditive ergodic theorem (Kingman) to sequences of displacement functions under random isometries.
• Leverage the Furstenberg-Lyons-Sullivan discretization to relate Brownian motion on Riemannian covers to random walks on the fundamental group.
• Construct a symmetric probability measure ν on the group Γ such that bounded harmonic functions on the cover correspond to harmonic functions on Γ.
• Use the existence of a first moment for ν and the properness of the group metric to ensure convergence of the random walk's linear drift.
• Prove that the rate of escape of Brownian motion vanishes a.s. if and only if the associated random walk has zero drift, linking stochastic and geometric properties.

Experimental results

Research questions

• RQ1 Under what conditions do products of random isometries on a proper metric space converge to a preferred direction in the metric compactification?
• RQ2 How does the subadditive ergodic theorem generalize to noncommutative settings involving isometries?
• RQ3 What is the relationship between the Liouville property of a Riemannian cover and the asymptotic behavior of Brownian motion on it?
• RQ4 Can the rate of escape of Brownian motion on a finite-volume Riemannian cover be characterized via discretization to a random walk on the fundamental group?
• RQ5 In which cases does the linear drift of a random walk on a locally compact group arise entirely from a character, given the Liouville property?

Key findings

• For a proper metric space, under integrable displacement, typical trajectories either exhibit sublinear diffusion or converge to a random geodesic ray in the metric compactification.
• The ray approximation property holds in CAT(0) spaces, where trajectories stay within O(1/n) distance of a random geodesic ray.
• When the space is Gromov hyperbolic, the theorem yields ergodic theorems for nonintegrable functions via different metrics on R.
• The Liouville property for a Riemannian cover of finite volume is equivalent to the almost sure vanishing of the linear drift of Brownian motion.
• The random walk associated to Brownian motion via discretization has a first moment and its drift is zero if and only if the Brownian motion has zero rate of escape.
• In the case of a non-amenable group, the cover is not Liouville, while for polycyclic groups, the cover is Liouville, confirming known results via the new framework.