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  • Journées équations aux dérivées partielles
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Tunnel effect for semiclassical random walk
Jean-François Bony1; Frédéric Hérau2; Laurent Michel3
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), article no. 6, 18 p.
  • 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.

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DOI: 10.5802/jedp.109
Author's affiliations:
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
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@incollection{JEDP_2014____A6_0,
     author = {Jean-Fran\c{c}ois Bony and Fr\'ed\'eric H\'erau and Laurent Michel},
     title = {Tunnel effect for semiclassical random walk},
     booktitle = {},
     series = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
     eid = {6},
     pages = {1--18},
     publisher = {Groupement de recherche 2434 du CNRS},
     year = {2014},
     doi = {10.5802/jedp.109},
     language = {en},
     url = {https://proceedings.centre-mersenne.org/articles/10.5802/jedp.109/}
}
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PY  - 2014
SP  - 1
EP  - 18
PB  - Groupement de recherche 2434 du CNRS
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DO  - 10.5802/jedp.109
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ER  - 
%0 Journal Article
%A Jean-François Bony
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%A Laurent Michel
%T Tunnel effect for semiclassical random walk
%J Journées équations aux dérivées partielles
%D 2014
%P 1-18
%I Groupement de recherche 2434 du CNRS
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%R 10.5802/jedp.109
%G en
%F JEDP_2014____A6_0
Jean-François Bony; Frédéric Hérau; Laurent Michel. Tunnel effect for semiclassical random walk. Journées équations aux dérivées partielles (2014), article  no. 6, 18 p. doi : 10.5802/jedp.109. https://proceedings.centre-mersenne.org/articles/10.5802/jedp.109/
  • References
  • Cited by

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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, Volume 10 (2011) no. 3, pp. 567-634 | MR | Zbl

[10] F. Hérau; M. Hitrik; J. Sjöstrand Supersymmetric structures for second order differential operators, Algebra i Analiz, Volume 25 (2013) no. 2, pp. 125-154 | MR

[11] T. Lelièvre; M. Rousset; G. Stoltz Free energy computations, Imperial College Press, 2010, pp. xiv+458 (A mathematical perspective) | MR | Zbl

[12] A. Martinez An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, 2002, pp. viii+190 | MR | Zbl

[13] A. Martinez; M. Rouleux Effet tunnel entre puits dégénérés, Comm. Partial Differential Equations, Volume 13 (1988) no. 9, pp. 1157-1187 | MR | Zbl

[14] M. Reed; B. Simon Methods of modern mathematical physics. IV. Analysis of operators, Academic Press, 1978, pp. xv+396 | MR | Zbl

[15] M. Zworski Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012, pp. xii+431 | MR | Zbl

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