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 nonlinear Langevin systems with single stable state perturbed by additive pure jump L\'evy noise small amplitude $\varepsilon>0$, where driving process is layered type. Under drift coercivity condition associated family processes $X^\varepsilon$ turns out to be exponentially ergodic equilibrium distribution $\mu^{\varepsilon}$ in total variation distance which extends result from Peng and Zhang (2018) arbitrary polynomial moments. The main results establish respect variation, under sufficient smoothing Blumenthal-Getoor index $\alpha>3/2$. That say, this setting we identify deterministic time scale $\mathfrak{t}_{\varepsilon}^{\mathrm{cut}}$ satisfying $\mathfrak{t}_ \varepsilon^{\mathrm{cut}} \rightarrow \infty$, as $\varepsilon 0$, respective window, $\mathfrak{t}_\varepsilon^{\mathrm{cut}} \pm o(\mathfrak{t}_\varepsilon^{\mathrm{cut}})$, during between current its essentially collapses $\varepsilon$ tends zero. In addition, extend dynamical characterization latter can described convergence such unique profile function first established Barrera Jara (2020) drift. leads conditions, verified examples, gradient subject symmetric $\alpha$-stable proof techniques differ completely Gaussian due absence Girsanov transforms couple equation linear approximation asymptotically even short times.