Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary
[Étude quantitative de la métastabilité des processus réversibles au moyen du complexe de Witten : le cas à bord.]
Mémoires de la Société Mathématique de France, no. 105 (2006) , 95 p.

Cet article prolonge des travaux antérieurs de Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein et Helffer-Klein-Nier. L’objet principal en est l’analyse des petites valeurs propres du Laplacien associé à la forme quadratique C 0 ( Ω ) v h 2 Ω v ( x ) 2 e - 2 f ( x ) / h d x , où Ω est un domaine borné régulier et f est une fonction de Morse sur M=Ω ¯. Les travaux précédents traitaient le cas d’une variété compacte M sans bord ou le cas M= n . Ici nous analysons le cas d’une variété compacte à bord. Après l’introduction d’un complexe de cohomologie de Witten adapté au cas à bord, nous donnons une description très précise des valeurs propres exponentiellement petites. En particulier, nous traitons l’effet du bord sur les développements asymptotiques.

This article is a continuation of previous works by Bovier-Eckhoff-Gayrard-Klein, Bovier-Gayrard-Klein and Helffer-Klein-Nier. The main object is the analysis of the small eigenvalues (as h0) of the Laplacian attached to the quadratic form C 0 ( Ω ) v h 2 Ω v ( x ) 2 e - 2 f ( x ) / h d x , where Ω is a bounded connected open set with C -boundary and f is a Morse function on M=Ω ¯. The previous works were devoted to the case of a manifold M which is compact but without boundary or n . Our aim is here to analyze the case with boundary. After the introduction of a Witten cohomology complex adapted to the case with boundary, we give a very accurate asymptotics for the exponentially small eigenvalues. In particular, we analyze the effect of the boundary in the asymptotics.

DOI : 10.24033/msmf.417
Classification : 58J10, 58J32, 58J65, 60J60, 81Q10, 81Q20
Keywords: Witten complex, Semiclassical expansion, exponentially small quantities, manifolds with boundary
Mot clés : Complexe de Witten, Développements semiclassiques, valeurs propres exponentiellement petites, variétés à bord
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Helffer, Bernard; Nier, Francis. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary. Mémoires de la Société Mathématique de France, Série 2, no. 105 (2006), 95 p. doi : 10.24033/msmf.417. http://numdam.org/item/MSMF_2006_2_105__1_0/

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