Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach.

Nier, Francis

Journées équations aux dérivées partielles (2004), article no. 8, 17 p.

We present here a simplified version of recent results obtained with B. Helffer and M. Klein. They are concerned with the exponentally small eigenvalues of the Witten Laplacian on $0$-forms. We show how the Witten complex structure is better taken into account by working with singular values. This provides a convenient framework to derive accurate approximations of the first eigenvalues of ${\Delta }_{f,h}^{\left(0\right)}$ and solves efficiently the question of weakly resonant wells.

[Bis] J.M. Bismut. The Witten complex and the degenerate Morse inequalities. J. Differ. Geom. 23, 207-240 (1986). | MR | Zbl

[BEGK] A. Bovier, M. Eckhoff, V. Gayrard, and M. Klein : Metastability in reversible diffusion processes I. Sharp asymptotics for capacities and exit times. Preprint 2002.

[BoGayKl] A. Bovier, V. Gayrard, and M. Klein. Metastability in reversible diffusion processes II Precise asymptotics for small eigenvalues. Preprint 2002 (new version in 2004). To appear in JEMS. | MR | Zbl

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

[CL] Kung Ching Chang, Jiaquan Liu. A cohomology complex for manifolds with boundary. Topological methods in non linear analysis. Volume 5, 1995, p. 325-340. | MR | Zbl

[CFKS] H.L Cycon, R.G Froese, W. Kirsch, and B. Simon. Schrödinger operators with application to quantum mechanics and global geometry. Text and Monographs in Physics. Springer-Verlag (1987). | MR | Zbl

[DiSj] M. Dimassi, J. Sjöstrand. Spectral Asymptotics in the semi-classical limit. London Mathematical Society. Lecture Note Series 268. Cambridge University Press (1999). | MR | Zbl

[Du] G.F.D. Duff. Differential forms in manifolds with boundary. Ann. of Math. 56 (1952), p. 115-127. | MR | Zbl

[DS] G.F.D. Duff, D.C. Spencer. Harmonic tensors on Riemannian manifolds with boundary. Ann. of Math. 56 (1952), p. 128-156. | MR | Zbl

[FrWe] M.I. Freidlin, A.D. Wentzell. Random perturbations of dynamical systems. Transl. from the Russian by Joseph Szuecs. 2nd ed. Grundlehren der Mathematischen Wissenschaften. 260. New York (1998). | MR | Zbl

[GoKr] I.C. Gohberg and M.G. Krejn. Introduction à la théorie des opérateurs linéaires non auto-adjoints dans un espace hilbertien. Monographies Universitaires de Mathématiques, No. 39. Dunod, Paris (1971). | MR

[Hel4] B. Helffer. Introduction to the semi-classical Analysis for the Schrödinger operator and applications. Springer Verlag. Lecture Notes in Math. n${}^{\circ }$1336 (1988). | MR | Zbl

[Hel] B. Helffer. Semi-classical analysis, Witten Laplacians and statistical mechanics. World Scientific (2002). | Zbl

[HKN] B. Helffer, M. Klein, and F. Nier. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach. Preprint 04-03, IRMAR, Univ. Rennes 1 (2004). | Zbl

[HelNi] B. Helffer and F. Nier. Hypoellipticity and spectral theory for Fokker-Planck operators and Witten Laplacians. Prépublication 03-25 de l’IRMAR, Univ. Rennes 1 (sept. 2003).

[HelNi3] B. Helffer and F. Nier. Quantitative analysis of metastability in reversible diffusion processes via a Witten complex approach : the case with boundary. In preparation.

[HelSj1] B. Helffer and J. Sjöstrand. Multiple wells in the semi-classical limit I, Comm. in PDE, 9(4), p. 337-408, (1984). | MR | Zbl

[HelSj2] B. Helffer and J. Sjöstrand. Puits multiples en limite semi-classique II -Interaction moléculaire-Symétries-Perturbations. Annales de l’IHP (section Physique théorique), Vol. 42, n${}^0$2, p. 127-212 (1985). | Numdam | MR | Zbl

[HelSj3] B. Helffer and J. Sjöstrand. Multiple wells in the semi-classical limit III. Math. Nachrichten 124, p. 263-313 (1985). | MR | Zbl

[HelSj4] B. Helffer and J. Sjöstrand. Puits multiples en limite semi-classique IV -Etude du complexe de Witten -. Comm. in PDE, 10(3), p. 245-340 (1985). | MR | Zbl

[HerNi] F. Hérau and F. Nier. Isotropic hypoellipticity and trend to the equilibrium for the Fokker-Planck equation with high degree potential. Arch. Ration. Mech. Anal. 171, n${}^0$ 2, 151–218 (2004). | MR | Zbl

[HolKusStr] R. Holley, S. Kusuoka, D. Stroock. Asymptotics of the spectral gap with applications to the theory of simulated annealing. J. Funct. Anal. 83, n${}^0$ 2, 333–347 (1989). | MR | Zbl

[Ko] V.N. Kolokoltsov. Semi-classical analysis for diffusions and stochastic processes. Lecture Notes in Mathematics 1724. Springer Verlag, Berlin 2000. | MR | Zbl

[Mic] L. Miclo. Comportement de spectres d’opérateurs à basse température. Bull. Sci. Math. 119, p. 529-533 (1995). | MR | Zbl

[Sim1] B. Simon. Trace ideals and their applications. Cambridge University Press IX, Lecture Notes Series vol. 35 (1979). | MR | Zbl

[Sim2] B. Simon. Semi-classical analysis of low lying eigenvalues, I.. Nondegenerate minima: Asymptotic expansions. Ann. Inst. Poincaré, 38, p. 296-307 (1983). | Numdam | MR | Zbl

[Schw] G. Schwarz. Hodge decomposition. A method for Solving Boundary Value Problems. Lect. Notes in Mathematics 1607, Springer (1995). | MR | Zbl

[Wit] E. Witten. Supersymmetry and Morse inequalities. J. Diff. Geom. 17, p. 661-692 (1982). | MR | Zbl

[Zh] Weiping Zhang. Lectures on Chern-Weil theory and Witten deformations. Nankai Tracts in Mathematics. Vol. 4. World Scientific (2002). | MR | Zbl