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

Abstract

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 Lévy noise of small amplitude $\mathit{\epsilon }>0$, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes $X^{\mathit{\epsilon }}$ turns out to be exponentially ergodic with equilibrium distribution ${\mathrm{\mu }}^{\mathit{\epsilon }}$ 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 $\mathit{\alpha }>\frac{3}{2}$. That is to say, in this setting we identify a deterministic time scale ${\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}}$ satisfying ${\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}}\to \mathrm{\infty }$, as $\mathit{\epsilon }\to 0$, and a respective time window, ${\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}}\pm o({\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}})$, during which the total variation distance between the current state and its equilibrium ${\mathrm{\mu }}^{\mathit{\epsilon }}$ essentially collapses as ε 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 Lévy 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 $\mathit{\alpha }>\frac{3}{2}$. 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.