Tunnel effect for semiclassical random walk
Journées équations aux dérivées partielles (2014), article no. 6, 18 p.

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.

DOI : 10.5802/jedp.109
Bony, Jean-François 1 ; Hérau, Frédéric 2 ; Michel, Laurent 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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Bony, Jean-François; Hérau, Frédéric; Michel, Laurent. Tunnel effect for semiclassical random walk. Journées équations aux dérivées partielles (2014), article  no. 6, 18 p. doi : 10.5802/jedp.109. http://www.numdam.org/articles/10.5802/jedp.109/

[1] Bony, J.-F.; Hérau, F.; Michel, L. Tunnel effect for semiclassical random walks (arXiv:1401.2935) | MR

[2] Bovier, A.; Gayrard, V.; Klein, M. Metastability in reversible diffusion processes. II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc., Volume 7 (2005) no. 1, pp. 69-99 | EuDML | MR | Zbl

[3] Cycon, H.; Froese, R.; Kirsch, W.; Simon, B. Schrödinger operators with application to quantum mechanics and global geometry, Texts and Monographs in Physics, Springer-Verlag, 1987, pp. x+319 | MR | Zbl

[4] Dimassi, M.; Sjöstrand, J. Spectral asymptotics in the semi-classical limit, London Mathematical Society Lecture Note Series, 268, Cambridge University Press, 1999, pp. xii+227 | MR | Zbl

[5] Helffer, B. Semi-classical analysis for the Schrödinger operator and applications, Lecture Notes in Mathematics, 1336, Springer-Verlag, 1988, pp. vi+107 | MR | Zbl

[6] Helffer, B.; Klein, M.; Nier, F. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp., Volume 26 (2004), pp. 41-85 | MR | Zbl

[7] Helffer, B.; Nier, F. Hypoelliptic estimates and spectral theory for Fokker-Planck operators and Witten Laplacians, Lecture Notes in Mathematics, 1862, Springer-Verlag, 2005, pp. x+209 | MR | Zbl

[8] Helffer, B.; Sjöstrand, J. Puits multiples en mécanique semi-classique. IV. Étude du complexe de Witten, Comm. Partial Differential Equations, Volume 10 (1985) no. 3, pp. 245-340 | MR | Zbl

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

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

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

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

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

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

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