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

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 Levy noise of small amplitude e>0, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes Xe turns out to be exponentially ergodic with equilibrium distribution μe in total variation distance which extends a result from [60] to arbitrary polynomial moments. The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index α>3 2. That is to say, in this setting we identify a deterministic time scale tecut satisfying tecut→∞, as e→0, and a respective time window, tecut±o(tecut), during which the total variation distance between the current state and its equilibrium μe essentially collapses as e 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 [9] to the Levy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric α-stable noise for α>3 2. The proof techniques differ completely from the Gaussian case due to the absence of a respective Girsanov transform which couples the nonlinear equation and the linear approximation asymptotically even for short times.