Metastable diffusions with degenerate drifts
[Diffusions métastables avec transports dégénérés]
Assal, Marouane1 ; Bony, Jean-François2 ; Michel, Laurent2
1 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las sophoras 173, Santiago (Chile)
2 IMB, UMR CNRS 5251, Université de Bordeaux, 33405 Talence (France)
Annales de l'Institut Fourier, Online first, 33 p.

Nous étudions le spectre du Laplacien de Witten semi-classique Δ f associé à une fonction lisse f sur ℝ d . Nous supposons que f est une fonction de Morse–Bott confinante. Sous cette hypothèse, nous montrons que Δ f admet des valeurs propres exponentiellement petites séparées du reste du spectre. De plus, nous établissons une formule d’Eyring–Kramers pour ces valeurs propres. Notre approche est basée sur des constructions microlocales de quasimodes au voisinage des sous-variétés critiques.

We study the spectrum of the semiclassical Witten Laplacian Δ f associated to a smooth function f on ℝ d . We assume that f is a confining Morse–Bott function. Under this assumption we show that Δ f admits exponentially small eigenvalues separated from the rest of the spectrum. Moreover, we establish Eyring–Kramers formula for these eigenvalues. Our approach is based on microlocal constructions of quasimodes near the critical submanifolds.

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DOI : 10.5802/aif.3636
Classification : 35J10, 35P20, 81Q20, 82C31
Keywords: Semiclassical analysis, Spectral asymptotic, Witten Laplacian, Overdamped Langevin dynamics.
Mot clés : Analyse semiclassique, asymptotique spectrale, laplacien de Witten, dynamique de Langevin suramortie.
Assal, Marouane 1 ; Bony, Jean-François 2 ; Michel, Laurent 2

1 Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las sophoras 173, Santiago (Chile)
2 IMB, UMR CNRS 5251, Université de Bordeaux, 33405 Talence (France) 
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Assal, Marouane; Bony, Jean-François; Michel, Laurent. Metastable diffusions with degenerate drifts. Annales de l'Institut Fourier, Online first, 33 p.

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