Tunnel effect for semiclassical random walk

Jean-François Bony[1]; Frédéric Hérau[2]; Laurent Michel[3]

  • [1] Institut Mathématiques de Bordeaux Université de Bordeaux, UMR CNRS 5251 351, cours de la Libération 33405 Talence Cedex, France
  • [2] Laboratoire de Mathématiques Jean Leray Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France
  • [3] Laboratoire Jean-Alexandre Dieudonné Université de Nice - Sophia Antipolis UMR CNRS 7351 06108 Nice Cedex 02, France

Journées Équations aux dérivées partielles (2014)

  • Volume: 8, Issue: 2, page 1-18
  • ISSN: 0752-0360

Abstract

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In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to 1 eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the spectral asymptotics based on some comparison argument.

How to cite

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Bony, Jean-François, Hérau, Frédéric, and Michel, Laurent. "Tunnel effect for semiclassical random walk." Journées Équations aux dérivées partielles 8.2 (2014): 1-18. <http://eudml.org/doc/275423>.

@article{Bony2014,
abstract = {In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to $1$ eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the spectral asymptotics based on some comparison argument.},
affiliation = {Institut Mathématiques de Bordeaux Université de Bordeaux, UMR CNRS 5251 351, cours de la Libération 33405 Talence Cedex, France; Laboratoire de Mathématiques Jean Leray Université de Nantes, UMR CNRS 6629 2, rue de la Houssinière 44322 Nantes Cedex 03, France; Laboratoire Jean-Alexandre Dieudonné Université de Nice - Sophia Antipolis UMR CNRS 7351 06108 Nice Cedex 02, France},
author = {Bony, Jean-François, Hérau, Frédéric, Michel, Laurent},
journal = {Journées Équations aux dérivées partielles},
keywords = {semiclassical random walks; tunnel effect; Markov transition kernel; pseudodifferential operators; spectral theory; Witten-Laplacian},
language = {eng},
number = {2},
pages = {1-18},
publisher = {Groupement de recherche 2434 du CNRS},
title = {Tunnel effect for semiclassical random walk},
url = {http://eudml.org/doc/275423},
volume = {8},
year = {2014},
}

TY - JOUR
AU - Bony, Jean-François
AU - Hérau, Frédéric
AU - Michel, Laurent
TI - Tunnel effect for semiclassical random walk
JO - Journées Équations aux dérivées partielles
PY - 2014
PB - Groupement de recherche 2434 du CNRS
VL - 8
IS - 2
SP - 1
EP - 18
AB - In this note we describe recent results on semiclassical random walk associated to a probability density which may also concentrate as the semiclassical parameter goes to zero. The main result gives a spectral asymptotics of the close to $1$ eigenvalues. This problem was studied in [1] and relies on a general factorization result for pseudo-differential operators. In this note we just sketch the proof of this second theorem. At the end of the note, using the factorization, we give a new proof of the spectral asymptotics based on some comparison argument.
LA - eng
KW - semiclassical random walks; tunnel effect; Markov transition kernel; pseudodifferential operators; spectral theory; Witten-Laplacian
UR - http://eudml.org/doc/275423
ER -

References

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  2. A. Bovier, V. Gayrard, M. Klein, Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc. 7 (2005), 69-99 Zbl1105.82025MR2120991
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  9. F. Hérau, M. Hitrik, J. Sjöstrand, Tunnel effect and symmetries for Kramers-Fokker-Planck type operators, J. Inst. Math. Jussieu 10 (2011), 567-634 Zbl1223.35246MR2806463
  10. F. Hérau, M. Hitrik, J. Sjöstrand, Supersymmetric structures for second order differential operators, Algebra i Analiz 25 (2013), 125-154 Zbl1303.81086MR3114853
  11. T. Lelièvre, M. Rousset, G. Stoltz, Free energy computations, (2010), Imperial College Press Zbl1227.82002MR2681239
  12. A. Martinez, An introduction to semiclassical and microlocal analysis, (2002), Springer-Verlag Zbl0994.35003MR1872698
  13. A. Martinez, M. Rouleux, Effet tunnel entre puits dégénérés, Comm. Partial Differential Equations 13 (1988), 1157-1187 Zbl0649.35073MR946285
  14. M. Reed, B. Simon, Methods of modern mathematical physics. IV. Analysis of operators, (1978), Academic Press Zbl0242.46001MR493421
  15. M. Zworski, Semiclassical analysis, 138 (2012), American Mathematical Society Zbl1252.58001MR2952218

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