Mathematics Colloquium
Speaker: Mayukh Mukherjee, IIT Bombay
Title: A \Delta-criterion, Green geometry and the strong Liouville property on groups
Time, day and date: 4:00:00 PM - 5:00:00 PM, Wednesday, October 29
Venue: Ramanujan Hall
Abstract: We develop a unified framework linking harmonic function growth, random-walk geometry, and heat kernel asymptotics on finitely generated groups. The central analytic device is a potential theory-based boundary functional whose vanishing characterises strong Liouville property, and we give general heat-kernel envelope conditions that force vanishing without requiring sharp two-sided estimates. This yields new, checkable criteria for strong Liouville on broad classes of groups (including polynomial growth and many subGaussian settings). In the opposite direction, we prove that on exponentially growing groups $\Delta$ does \emph{not} decay on balls (under mild on-diagonal bounds), forcing the existence of non-constant positive harmonic functions. On the geometric side, we show that a trivial Martin boundary collapses the Green geometry and the Green speed vanishes along any path with finite word-speed. Two speed results refine the picture on nilpotent groups. First, there are natural heavy-tail regimes with speed vanishing in probability. Second, under a uniform cone lower bound on jump directions, any positive word-speed implies finite first moment - and by symmetry the drift is then zero almost surely.

This is based on joint work with Soumyadeb Samanta and Soumyadip Thandar