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Tunnel effect for Kramers-Fokker-Planck type operators. (English) Zbl 1141.82011
This work is a continuation of [F. Hérau, J. Sjöstrand and C. C. Stolk, Commun. Partial Differ. Equations 30, No. 5–6, 689–760 (2005; Zbl 1083.35149)]. The authors consider operators of Kramers-Fokker-Planck type of the form \[ P=y\cdot h\partial_x - V' (x)\cdot h \partial_y +\frac{\gamma}{2} (-h\partial_y+y) \cdot (h\partial_y+y) \qquad x,y\in \mathbb{R}^n \] in the semi-classical limit \(h\rightarrow 0\) corresponding to low temperature, such that the exponent of the associated Maxwellian is a Morse function with two local minima and a saddle point. Under suitable additional assumptions they establish the complete asymptotics of the exponentially small splitting between the first two eigenvalues.

MSC:
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
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