[Analyse spectrale d’une marche aléatoire semiclassique associée à un potentiel confinant général]
We consider a semiclassical random walk with respect to a probability measure associated to a potential with a finite number of critical points. We recover the spectral results from [1] on the corresponding operator in a more general setting and with improved accuracy. In particular we do not make any assumption on the distribution of the critical points of the potential, in the spirit of [14]. Our approach consists in adapting the ideas from [14] to the recent gaussian quasimodes framework which appears to be more robust than the usual methods, especially when dealing with non local operators.
On considère une marche aléatoire semiclassique définie à l’aide d’une mesure de probabilité associée à un potentiel présentant un nombre fini de points critiques. On retrouve les résultats de [1] sur le spectre de l’opérateur associé dans un contexte plus général et avec une précision améliorée. On ne formule en particulier aucune hypothèse sur la distribution des points critiques du potentiel, dans l’esprit de [14]. Notre approche consiste à adapter les idées de [14] aux récentes techniques de quasimodes guassiens qui s’avèrent être plus robustes que les méthodes usuelles, en particulier pour l’étude d’opérateurs non locaux.
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Keywords: analysis of PDEs, probability, spectral theory
Mots-clés : analyse des EDPs, probabilités, théorie spectrale
Thomas Normand 1
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@article{AFST_2025_6_34_5_1259_0,
author = {Thomas Normand},
title = {Spectral analysis of a semiclassical random walk associated to a general confining potential},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {1259--1294},
year = {2025},
publisher = {Universit\'e de Toulouse, Toulouse},
volume = {Ser. 6, 34},
number = {5},
doi = {10.5802/afst.1832},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1832/}
}
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Thomas Normand. Spectral analysis of a semiclassical random walk associated to a general confining potential. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 34 (2025) no. 5, pp. 1259-1294. doi: 10.5802/afst.1832
[1] Tunnel effect for semiclassical random walks, Anal. PDE, Volume 8 (2015) no. 2, pp. 289-332 | DOI | Zbl | MR
[2] Eyring–Kramers law for Fokker–Planck type differential operators (2022) | arXiv
[3] Metastability In Reversible Diffusion Processes I. Sharp Asymptotics For Capacities And Exit Times, J. Eur. Math. Soc., Volume 6 (2004) no. 4, pp. 399-424 | DOI | Zbl | MR
[4] Metastability in reversible diffusion processes II. Precise asymptotics for small eigenvalues, J. Eur. Math. Soc., Volume 7 (2005) no. 1, pp. 69-99 | DOI | Zbl | MR
[5] Micro-local analysis for the Metropolis algorithm, Math. Z., Volume 262 (2009) no. 2, pp. 411-447 | DOI | Zbl | MR
[6] Geometric Analysis of the Metropolis Algorithm in Lipschitz Domain, Invent. Math., Volume 185 (2011) no. 2, pp. 239-281 | Zbl | DOI
[7] Spectral asymptotics in the semiclassical limit, London Mathematical Society Lecture Note Series, 268, London Mathematical Society, 1999 | MR | Zbl | DOI
[8] Spectral analysis of random walk operators on euclidian space, Math. Res. Lett., Volume 18 (2011) no. 3, pp. 405-424 | DOI | Zbl | MR
[9] Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach, Mat. Contemp., Volume 26 (2004), pp. 41-85 | Zbl | MR
[10] Tunnel effect for Kramers–Fokker–Planck type operators, Ann. Henri Poincaré, Volume 9 (2008) no. 2, pp. 209-274 | DOI | Zbl
[11] Tunnel effect and symmetries for Kramers–Fokker–Planck type operators, J. Inst. Math. Jussieu, Volume 10 (2011) no. 3, pp. 567-634 | MR | DOI | Zbl
[12] Semi-classical analysis of a random walk on a manifold, Ann. Probab., Volume 38 (2010) no. 1, pp. 277-315 | MR | DOI | Zbl
[13] Free Energy Computations. A Mathematical Perspective, Imperial College Press, 2010 | MR | DOI | Zbl
[14] About small eigenvalues of Witten Laplacian, Pure Appl. Anal., Volume 1 (2019) no. 2, pp. 149-206 | MR | DOI | Zbl
[15] Agmon-type exponential decay estimates for pseudo-differential operators, J. Math. Sci. Univ. Tokyo, Volume 5 (1998) no. 4, pp. 693-712 | MR | Zbl
[16] Metastability results for a class of linear Boltzmann equations, Ann. Henri Poincaré, Volume 24 (2023) no. 11, pp. 4013-4067 | DOI | Zbl | MR
[17] Spectral asymptotics and metastability for the linear relaxation Boltzmann equation, J. Spectr. Theory, Volume 14 (2024) no. 3, pp. 1195-1242 | DOI | Zbl | MR
[18] Sharp spectral asymptotics for non-reversible metastable diffusion processes, Probab. Math. Phys., Volume 1 (2020) no. 1, pp. 3-53 | DOI | Zbl | MR
[19] Semiclassical analysis, Graduate Studies in Mathematics, 138, American Mathematical Society, 2012 | Zbl | DOI | MR
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