SEMINAIRE : Spectral analysis of Fokker-Planck and Kramers-Fokker-Planck operators

Spectral analysis of Fokker-Planck

and Kramers-Fokker-Planck operators

Francis Nier

IRMAR, UMR-CNRS 6625, Univ. Rennes 1

INRIA-MICMAC project-team

In the last ten years, many advances have be done in the spectral analysis of

the Fokker-Planck operator

2∇V (x).∇x −

1

β

∆x

and the Kramers-Fokker-Planck operators

v.∂x −

1

m

(∂xV (x)).∂b +

γ

mβ

(−∂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 =

1

kBT

, 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

boundary.

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

−tL

k ≤ Cαe

−αt

• 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)

Sources : 

http://cermics.enpc.fr/seminaires/cs/Seminaire11-12/nier....

http://cermics.enpc.fr/seminaires/cs/seminaire.html