Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary

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.]

Helffer, Bernard ; Nier, Francis

Mémoires de la Société Mathématique de France, no. 105 (2006) , 95 p.

Résumé
Abstract

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 $C_{0}^{\infty} (Ω) ∋ v \mapsto h^{2} \int_{Ω} {|\nabla v (x)|}^{2} e^{- 2 f (x) / h} d x$ , où $Ω$ est un domaine borné régulier et $f$ est une fonction de Morse sur $M = \bar{Ω}$ . Les travaux précédents traitaient le cas d’une variété compacte $M$ sans bord ou le cas $M = ℝ^{n}$ . 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 $h \to 0$ ) of the Laplacian attached to the quadratic form $C_{0}^{\infty} (Ω) ∋ v \mapsto h^{2} \int_{Ω} {|\nabla v (x)|}^{2} e^{- 2 f (x) / h} d x$ , where $Ω$ is a bounded connected open set with $C^{\infty}$ -boundary and $f$ is a Morse function on $M = \bar{Ω}$ . The previous works were devoted to the case of a manifold $M$ which is compact but without boundary or $ℝ^{n}$ . 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.

MR Zbl | 2 citations dans Numdam

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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