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2021 The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise
Gerardo Barrera, Michael A. Högele, Juan Carlos Pardo
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Electron. J. Probab. 26: 1-76 (2021). DOI: 10.1214/21-EJP685

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 ε>0, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes Xε turns out to be exponentially ergodic with equilibrium distribution με 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 tεcut satisfying tεcut, as ε0, and a respective time window, tεcut±o(tεcut), during which the total variation distance between the current state and its equilibrium με 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 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.

Funding Statement

Supported by the Academy of Finland, Facultad de Ciencias at Universidad de los Andes and CONACyT-MEXICO.

Acknowledgments

The research of GBV has been supported by the Academy of Finland, via the Matter and Materials Profi4 university profiling action. GBV also would like to express his gratitude to University of Helsinki for all the facilities used along the realization of this work. The research of MAH has been supported by the Proyecto de la Convocatoria 2020–2021: “Stochastic dynamics of systems perturbed with small Markovian noise with applications in biophysics, climatology and statistics” of the School of Sciences (Facultad de Ciencias) at Universidad de los Andes. JCP acknowledges support from CONACyT-MEXICO CB-250590. The authors would like to thank professor M. Jara and professor R. Imbuzeiro Oliveira both at IMPA for ideas how to construct the example given in Subsubsection 1.3.5.

Citation

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Gerardo Barrera. Michael A. Högele. Juan Carlos Pardo. "The cutoff phenomenon in total variation for nonlinear Langevin systems with small layered stable noise." Electron. J. Probab. 26 1 - 76, 2021. https://doi.org/10.1214/21-EJP685

Information

Received: 12 December 2020; Accepted: 10 August 2021; Published: 2021
First available in Project Euclid: 20 September 2021

Digital Object Identifier: 10.1214/21-EJP685

Subjects:
Primary: 37A25, 37A30, 60F05, 60G51, 60G52, 65C30

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