Abstract
A particle subject to a white noise external forcing moves according to a Langevin process. Consider now that the particle is reflected at a boundary which restores a portion c of the incoming speed at each bounce. For c strictly smaller than the critical value \(c_{\mathit{crit}} =\exp (-\pi /\sqrt{3})\), the bounces of the reflected process accumulate in a finite time, yielding a very different behavior from the most studied cases of perfectly elastic reflection—c = 1—and totally inelastic reflection—c = 0. We show that nonetheless the particle is not necessarily absorbed after this accumulation of bounces. We define a “resurrected” reflected process as a recurrent extension of the absorbed process, and study some of its properties. We also prove that this resurrected reflected process is the unique solution to the stochastic partial differential equation describing the model, for which well-posedness is nothing obvious. Our approach consists in defining the process conditioned on never being absorbed, via an h-transform, and then giving the Itō excursion measure of the recurrent extension thanks to a formula fairly similar to Imhof’s relation.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
P. Ballard, The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154, 199–274 (2000)
J. Bect, Processus de Markov diffusifs par morceaux: outils analytiques et numriques. PhD thesis, Supelec, 2007
J. Bertoin, Reflecting a Langevin process at an absorbing boundary. Ann. Probab. 35(6), 2021–2037 (2007)
J. Bertoin, A second order SDE for the Langevin process reflected at a completely inelastic boundary. J. Eur. Math. Soc. 10(3), 625–639 (2008)
R.M. Blumenthal, On construction of Markov processes. Z. Wahrsch. Verw. Gebiete 63(4), 433–444 (1983)
M. Bossy, J.-F. Jabir, On confined McKean Langevin processes satisfying the mean no-permeability boundary condition. Stochastic Process. Appl. 121, 2751–2775 (2011)
A. Bressan, Incompatibilit dei teoremi di esistenza e di unicit del moto per un tipo molto comune e regolare di sistemi meccanici. Ann. Scuola Norm. Sup. Pisa Serie III 14, 333–348(1960)
C.M. Goldie, Implicit renewal theory and tails of solutions of random equations. Ann. Appl. Probab. 1(1), 126–166 (1991)
J.P. Gor′kov, A formula for the solution of a certain boundary value problem for the stationary equation of Brownian motion. Dokl. Akad. Nauk SSSR 223(3), 525–528 (1975)
J.-P. Imhof, Density factorizations for Brownian motion, meander and the three-dimensional Bessel process, and applications. J. Appl. Probab. 21(3), 500–510 (1984)
E. Jacob, Excursions of the integral of the Brownian motion. Ann. Inst. H. Poincaré Probab. Stat. 46(3), 869–887 (2010)
E. Jacob, Langevin process reflected on a partially elastic boundary I. Stoch. Process. Appl. 122(1), 191–216 (2012)
H. Kesten, Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207–248 (1973)
A. Lachal, Les temps de passage successifs de l’intégrale du mouvement brownien. Ann. Inst. H. Poincaré Probab. Stat. 33(1), 1–36 (1997)
A. Lachal, Application de la théorie des excursions à l’intégrale du mouvement brownien. In Séminaire de Probabilités XXXVII, vol. 1832 of Lecture Notes in Math. (Springer, Berlin, 2003), pp. 109–195
B. Maury, Direct simulation of aggregation phenomena. Comm. Math. Sci. 2(suppl. 1), 1–11 (2004)
H.P. McKean, Jr., A winding problem for a resonator driven by a white noise. J. Math. Kyoto Univ. 2, 227–235 (1963)
P.E. Protter, Stochastic integration and differential equations, vol. 21 of Stochastic Modelling and Applied Probability, 2nd edn, Version 2.1, Corrected third printing (Springer, Berlin, 2005)
V. Rivero, Recurrent extensions of self-similar Markov processes and Cramér’s condition. Bernoulli 11(3), 471–509 (2005)
M. Schatzman, Uniqueness and continuous dependence on data for one dimensional impact problems. Math. Comput. Model. 28, 1–18 (1998)
S.J. Taylor, J.G. Wendel, The exact Hausdorff measure of the zero set of a stable process. Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 6, 170–180 (1966)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Jacob, E. (2013). Langevin Process Reflected on a Partially Elastic Boundary II. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités XLV. Lecture Notes in Mathematics(), vol 2078. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00321-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-00321-4_9
Published:
Publisher Name: Springer, Heidelberg
Print ISBN: 978-3-319-00320-7
Online ISBN: 978-3-319-00321-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)