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 L\'evy noise of small amplitude $\varepsilon>0$, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes $X^\varepsilon$ turns out to be exponentially ergodic with equilibrium distribution $\mu^{\varepsilon}$ in total variation distance which extends a result from Peng and Zhang (2018) 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 $\alpha>3/2$. That is to say, in this setting we identify a deterministic time scale $\mathfrak{t}_{\varepsilon}^{\mathrm{cut}}$ satisfying $\mathfrak{t}_ \varepsilon^{\mathrm{cut}} \rightarrow \infty$, as $\varepsilon \rightarrow 0$, and a respective time window, $\mathfrak{t}_\varepsilon^{\mathrm{cut}} \pm o(\mathfrak{t}_\varepsilon^{\mathrm{cut}})$, during which the total variation distance between the current state and its equilibrium $\mu^{\varepsilon}$ essentially collapses as $\varepsilon$ 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 $\alpha>3/2$. The proof techniques differ completely from the Gaussian case due to the absence of respective Girsanov transforms which couple the nonlinear equation and the linear approximation asymptotically even for short times.