SEMINAIRE : Spectral analysis of Fokker-Planck and Kramers-Fokker-Planck operators
Spectral analysis of Fokker-Planck
and Kramers-Fokker-Planck operators
IRMAR, UMR-CNRS 6625, Univ. Rennes 1
In the last ten years, many advances have be done in the spectral analysis of
the Fokker-Planck operator
2∇V (x).∇x −
and the Kramers-Fokker-Planck operators
(∂xV (x)).∂b +
(−∂v + mβv).∂v .
The main questions which were studied were about estimating the rate of
return to the equilibrium, w.r.t the mass m, the friction coeﬃcient γ and the
inverse temperature β =
, and more generally the life time of metastable
states in the small temperature asymptotics (large β). These operators can
be considered on the whole space, on riemannian manifolds or domains with
Mathematically this has connection with hypoellipticity, ﬁne spectral analysis of non self-adjoint operators, accurate estimate of the tunnel eﬀect for
Schr¨odinger type operators, diﬀerential geometry, and even algebraic topology or Lie algebras.
In this short course, I will present the main ideas, illustrated by simple examples and give some general results which can be useful in some other
frameworks. Many references to recent (and less recent) works will be given
and brieﬂy explained.
The outline of the course will be:
• Introduction: Smoluchowski, Langevin processes and their semigroup.
• Semigroups, contour integrals and hypoellipticity.
• Results about the Kramers-Fokker-Planck equation and its relation
with the Fokker-Planck equation.
• About the constant Cα for semigroup decay estimates ke
k ≤ Cαe
• Additional structures: a) PT-symmetry; b) supersymmetry; c) nilpotent Lie algebras and maximal hypoellipticity.
Francis Nier (Université de Rennes)
lundi 7 novembre 2011 à 14h00 (B413)
mardi 15 novembre 2011 à 14h00 (la salle sera annoncée ultérieurement)
lundi 21 novembre 2011 à 14h00 (B413)
lundi 5 décembre 2011 à 14h00 (B413)