Session S04 - Random Walks and Related Topics

Monday, July 19, 17:20 ~ 17:50 UTC-3

The cutoff phenomenon in total variation for nonlinear Langevin systems with stable type noise

Juan Carlos Pardo

Centro de Investigación en Matemáticas, Mexico   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump L\'evy noise of small amplitude, where the driving noise process is of layered stable type.

Under a drift coercivity condition the associated family of solutions turns out to be exponentially ergodic with equilibrium distribution in total variation distance which extends a result from Peng and Zhang (2018) to arbitrary moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index larger than 3/2. That is to say, in this setting we identify a deterministic time scale which tends to infinity in the limit of small noise and a respective time window during which the total variation distance between the current state and its equilibrium essentially collapses as the noise amplitude tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in Barrera and Jara (2020) to the L\'evy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric alpha-stable noise for index larger than 3/2.

Joint work with Gerardo Barrera (University of Helsinki, Finland) and Michael Hoegele (Universidad de los Andes. Bogot\'a, Colombia).

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