Dynamical systems, ergodic theory, probability theory, and their applications to physical systems — diffusion, transport, and statistical mechanics. A particular focus on infinite-measure settings and random environments.
My research sits at the meeting point of dynamical systems, ergodic theory, and probability theory, with a unifying interest in the long-time statistical behavior of deterministic and stochastic systems that live in infinite or non-compact spaces.
A central object in my work is the billiard: a point particle bouncing inside a domain. In compact, dispersing billiards (Sinai billiards), ergodicity and mixing are classical results. My focus is on non-compact or aperiodic billiard tables — infinite-step billiards, semi-dispersing cusps, internal-wave billiards in trapezoids, Lorentz gases with infinite horizon — where escape to infinity competes with recurrence, and where standard finite-measure tools fail entirely. A key question is: does the billiard flow return to any region? Is it ergodic? Does it exhibit anomalous diffusion?
A second major thread is infinite-measure ergodic theory. Standard mixing is not meaningful for systems with infinite invariant measure: correlation functions do not decay to zero in a useful sense. I introduced and developed the notions of global-local mixing and global observables — functions that have a well-defined infinite-volume average — which allow one to formulate and prove physically relevant decorrelation properties for Pomeau–Manneville maps, Boole maps, uniformly expanding Markov maps, random walks, and other infinite-measure systems.
A third theme is random walks in disordered (Lévy) environments. In these models, the random medium has heavy-tailed gaps, leading to anomalous transport: the walker cannot be described by standard Brownian scaling. With collaborators I have proved quenched and annealed limit theorems, laws of large numbers, and large-deviation results in one dimension, as well as studied Lévy flights on Lévy random media and their connections to multilayer stochastic models of human mobility.
More recently I have become an active advocate for formal mathematics and proof assistants, particularly Lean 4 and Mathlib. I believe the formalization of results in ergodic theory and dynamical systems is both feasible and important, and I am directly involved in organizing events to promote this direction in Italy and internationally.