Research and Activities
My research lies at the interface of stochastic processes, dynamical systems, and mathematical physics. A central theme of my work is the rigorous analysis of abrupt convergence to equilibrium, commonly referred to as the cut-off phenomenon, for random perturbations of deterministic dynamical systems driven by various types of noises, including Brownian motion, fractional Brownian motion and Lévy processes. In particular, I investigate the small-temperature asymptotic regime for stochastic differential equations, where the convergence to the dynamical equilibrium may occur over a time window relative small to the natural relaxation scale.
The cut-off phenomenon was originally identified and systematically studied in the 1980s in the context of classical probabilistic models such as card-shuffling schemes, Ehrenfest urn models, and random transpositions. From a mathematical standpoint, determining whether a given family of stochastic processes exhibits a cut-off is highly nontrivial: it typically requires a detailed, quantitative understanding of the full transient and asymptotic dynamics of the process, rather than merely its long-time behavior.
Beyond this core theme, my research encompasses:
Random polynomials and random structured matrices, including circulant and Toeplitz ensembles.
Stability of trinomials with general complex coefficients.
Root-counting problems for harmonic trinomials.
Ergodicity for locally monotone SPDEs.
Stability theory for multivariate stochastic linear systems.
Ergodic decomposition of the velocity-flip model arising in nonequilibrium statistical mechanics.