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A Strong Averaging Principle for Lévy Diffusions in Foliated Spaces with Unbounded Leaves

  • Paulo Henrique da Costa
  • Michael A. HögeleEmail author
  • Paulo Regis Ruffino
Chapter
Part of the Understanding Complex Systems book series (UCS)

Abstract

This article extends a strong averaging principle for Lévy diffusions which live on the leaves of a foliated manifold subject to small transversal Lévy type perturbation to the case of non-compact leaves. The main result states that the existence of p-th moments of the foliated Lévy diffusion for \(p\geqslant 2\) and an ergodic convergence of its coefficients in Lp implies the strong Lp convergence of the fast perturbed motion on the time scale tε to the system driven by the averaged coefficients. In order to compensate the non-compactness of the leaves we use an estimate of the dynamical system for each of the increments of the canonical Marcus equation derived in da Costa and Högele (Potential Anal 47(3):277–311, 2017), the boundedness of the coefficients in Lp and a nonlinear Gronwall-Bihari type estimate. The price for the non-compactness are slower rates of convergence, given as p-dependent powers of ε strictly smaller than 1∕4.

Notes

Acknowledgements

The author PHC would like to thank the Department of Mathematics of Brasilia University for providing support. The authors MAH and PRR would like express his gratitude for the hospitality received at the Departameto de Matemática at Universidade de Brasília and the IMECC at UNICAMP in February 2018. The funding of MAH by the FAPA project “Stochastic dynamics of Lévy driven systems” at the School of Science at Universidad de los Andes is greatly acknowledged. The author PRR is partially supported by Brazilian CNPq proc. nr. 305462/2016-4, by FAPESP proc. nr. 2015/07278-0 and 2015/50122-0.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Paulo Henrique da Costa
    • 1
  • Michael A. Högele
    • 2
    Email author
  • Paulo Regis Ruffino
    • 3
  1. 1.Departamento de MatemáticaUniversidade de BrasíliaBrasíliaBrazil
  2. 2.Departamento de MatemáticasUniversidad de los AndesBogotáColombia
  3. 3.IMECCUniversidade Estadual de CampinasCampinasBrazil

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