Space-time Martin boundary and ratio-limit boundaries

Ratio-limit boundaries were first studied for their applications to Toeplitz C-algebras of random walk, but are also interesting in their own right for measuring new types of behavior at infinity. For the purpose of describing Toeplitz C-algebras of random walks, new boundaries need to be identified in more precise terms. One such boundary is the so-called space-time Martin boundary, as studied by Lalley for random walks on the free group.

In this talk we will discuss ratio-limit boundaries and some work in progress on space-time Martin boundaries of random walks on discrete groups. The space-time Martin boundary is related to the notion of stability studied by Picardello and Woess, which elucidates potential descriptions of the space-time Martin boundaries for random walks on \mathbb{Z}^d and on hyperbolic groups.