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

@book{MSMF_2006_2_105__1_0,
     author = {Helffer, Bernard and Nier, Francis},
     title = {Quantitative analysis of metastability in reversible diffusion processes via a {Witten} complex approach: the case with boundary},
     series = {M\'emoires de la Soci\'et\'e Math\'ematique de France},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {105},
     year = {2006},
     doi = {10.24033/msmf.417},
     mrnumber = {2270650},
     zbl = {1108.58018},
     language = {en},
     url = {http://www.numdam.org/item/MSMF_2006_2_105__1_0/}
}

TY  - BOOK
AU  - Helffer, Bernard
AU  - Nier, Francis
TI  - Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary
T3  - Mémoires de la Société Mathématique de France
PY  - 2006
IS  - 105
PB  - Société mathématique de France
UR  - http://www.numdam.org/item/MSMF_2006_2_105__1_0/
DO  - 10.24033/msmf.417
LA  - en
ID  - MSMF_2006_2_105__1_0
ER  -

%0 Book
%A Helffer, Bernard
%A Nier, Francis
%T Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach: the case with boundary
%S Mémoires de la Société Mathématique de France
%D 2006
%N 105
%I Société mathématique de France
%U http://www.numdam.org/item/MSMF_2006_2_105__1_0/
%R 10.24033/msmf.417
%G en
%F MSMF_2006_2_105__1_0

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/

Bibliographie
Cité par

[1] J. Bismut – « The Witten complex and the degenerate Morse inequalities », J. Differ. Geom. 23 (1986), p. 207–240. | MR | Zbl

[2] C. Bolley & B. Helffer – « Application of semi-classical analysis to the asymptotic study of the supercooling field of a superconducting material », Ann. Inst. H. Poincaré Phys. Théor. 58 (1991), no. 2, p. 189–233. | MR | EuDML | Zbl | Numdam

[3] A. Bovier, M. Eckhoff, V. Gayrard & M. Klein – « Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times », JEMS 6 (2004), no. 4, p. 399–424. | MR | EuDML | Zbl

[4] A. Bovier, V. Gayrard & M. Klein – « Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues », JEMS 7 (2004), no. 1, p. 69–99. | MR | EuDML | Zbl

[5] D. Burghelea – « Lectures on Witten-Helffer-Sjöstrand theory », Gen. Math. 5 (1997), p. 85–99. | MR | EuDML | Zbl

[6] K. Chang & J. Liu – « A cohomology complex for manifolds with boundary », Topol. Methods in Nonlinear Anal. 5 (1995), no. 2, p. 325–340. | MR | Zbl

[7] H. Cycon, R. Froese, W. Kirsch & B. Simon – Schrödinger Operators with Application to Quantum Mechanics and Global Geometry, Text and Monographs in Physics, Springer-Verlag, 1987. | MR | Zbl

[8] M. Dauge – Elliptic boundary value problems on corner domains. Smoothness and asymptotic solutions, Lect. Notes in Math., vol. 1341, Springer Verlag, 1988. | MR | Zbl

[9] M. Dimassi & J. Sjöstrand – Spectral asymptotics in the semiclassical limit, London Mathematical Society Lecture Note Series, vol. 268, Cambridge University Press, 1999. | MR

[10] M. Freidlin & A. Wentzell – Random perturbations of dynamical systems. 2nd edition, 2nd éd., Grundlehren der Mathematischen Wissenschaften, vol. 260, Springer Verlag, 1998. | MR

[11] S. Gallot, D. Hullin & J. Lafontaine – Riemannian Geometry, 2nd éd., Universitext, Springer Verlag, 1993. | MR | Zbl

[12] I. Gohberg & M. Krejn – Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien, Monographies Universitaires de Mathématiques, vol. 39, Dunod, 1971. | MR

[13] P. Grisvard – Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman, 1985. | MR | Zbl

[14] P. Guérini – « Prescription du spectre du Laplacien de Hodge-de Rham », Ann. ENS 37 (2004), no. 2, p. 270–303. | MR | EuDML | Zbl | Numdam

[15] B. Helffer – « Etude du laplacien de Witten associé à une fonction de Morse dégénérée », Publications de l’université de Nantes, Séminaire EDP 1987-88.

[16] —, Semi-classical analysis for the Schrödinger operator and applications, Lect. Notes in Mathematics, vol. 1336, Springer Verlag, Berlin, 1988.

[17] —, Semi-classical analysis, Witten Laplacians and statistical mechanics, World Scientific, 2002.

[18] B. Helffer, M. Klein & F. Nier – « Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. », Matematica Contemporanea 26 (2004), p. 41–85. | MR | Zbl

[19] B. Helffer & J. Sjöstrand – « Multiple wells in the semi-classical limit I », Comm. Partial Differential Equations. 9 (1984), no. 4, p. 337–408. | MR | Zbl

[20] —, « Puits multiples en limite semi-classique II-interaction moléculaire-symétries-perturbations », Ann. Inst. H. Poincaré Phys. Théor. 42 (1985), no. 2, p. 127–212. | MR | EuDML | Zbl | Numdam

[21] —, « Puits multiples en limite semi-classique IV-étude du complexe de Witten - », Comm. Partial Differential Equations 10 (1985), no. 3, p. 245–340. | MR | Zbl

[22] —, « Puits multiples en limite semi-classique V - Etude des minipuits - », in Current topics in partial differential equations, 1986. | Zbl

[23] R. Holley, R. Kusuoka & D. Stroock – « Asymptotics of the spectral gap with applications to the theory of simulated annealing », J. Funct. Anal. 83 (1989), no. 2, p. 333–347. | MR | Zbl

[24] J. Johnsen – « On the spectral properties of Witten Laplacians, their range projections and Brascamp-Lieb’s inequality », Integral Equations and Operator Theory 36 (2000), no. 2, p. 288–324. | MR | Zbl

[25] V. Kolokoltsov – Semiclassical analysis for diffusions and stochastic processes, Lect. Notes in Mathematics, vol. 1724, Springer Verlag, 2000. | MR | Zbl

[26] V. Kolokoltsov & K. Makarov – « Asymptotic spectral analysis of a small diffusion operator and the life times of the corresponding diffusion process », Russian J. Math. Phys. 4 (1996), no. 3, p. 341–360. | MR | Zbl

[27] V. Kondratiev – « Boundary value problems for elliptic equations in domains with conical or angular points », Trudy Moskov. Mat. Obshch. 16 (1967), p. 209–292. | MR

[28] L. Miclo – « Comportement de spectres d’opérateurs à basse température », Bull. Sci. Math. 119 (1995), p. 529–533. | MR

[29] G. Schwarz – Hodge decomposition. A method for Solving Boundary Value Problems, Lect. Notes Series, vol. 1607, Springer Verlag, 1995. | MR

[30] C. Simader – « Essential self-adjointness of schrödinger operators bounded from below », Math. Z. 159 (1978), p. 47–50. | MR | EuDML | Zbl

[31] B. Simon – Trace ideals and their applications., Lecture Note Series, vol. 35, London Mathematical Society, Cambridge University Press. IX, 1979. | MR | Zbl

[32] —, « Semi-classical analysis of low lying eigenvalues, I. Nondegenerate minima: Asymptotic expansions », Ann. Inst. H. Poincaré Phys. Théor. 38 (1983), p. 296–307. | EuDML

[33] E. Witten – « Supersymmetry and Morse inequalities », J. Differ. Geom. 17 (1982), p. 661–692. | MR | Zbl

[34] W. Zhang – Lectures on Chern-Weil theory and Witten deformations, Nankai Tracts in Mathematics, vol. 4, World Scientific, 2002. | MR

Cité par Sources :