8 research outputs found
Short and long time behavior of the Fokker-Planck equation in a confining potential and applications
We consider the linear Fokker-Planck equation in a confining potential in
space dimension . Using spectral methods, we prove bounds on the
derivatives of the solution for short and long time, and give some
applications.Comment: corrected version of the outdated article "Uniform bounds and
exponential time decay results for the Vlasov-Poisson-Fokker-Planck system
Anisotropic hypoelliptic estimates for Landau-type operators
We establish global hypoelliptic estimates for linear Landau-type operators.
Linear Landau-type equations are a class of inhomogeneous kinetic equations
with anisotropic diffusion whose study is motivated by the linearization of the
Landau equation near the Maxwellian distribution. By introducing a microlocal
method by multiplier which can be adapted to various hypoelliptic kinetic
equations, we establish for linear Landau-type operators optimal global
hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is
exactly related to the anisotropy of the diffusion.Comment: 44 page
Tunnel effect for Kramers-Fokker-Planck type operators
We consider operators of Kramers-Fokker-Planck type in the semi-classical
limit such that the exponent of the associated Maxwellian is a Morse function
with two local minima and a saddle point. Under suitable additional assumptions
we establish the complete asymptotics of the exponentially small splitting
between the first two eigenvalues.Comment: 77 page
Tunnel effect for Kramers-Fokker-Planck type operators: return to equilibrium and applications
In the first part of this work, we consider second order supersymmetric
differential operators in the semiclassical limit, including the
Kramers-Fokker-Planck operator, such that the exponent of the associated
Maxwellian is a Morse function with two local minima and one saddle
point. Under suitable additional assumptions of dynamical nature, we establish
the long time convergence to the equilibrium for the associated heat semigroup,
with the rate given by the first non-vanishing, exponentially small,
eigenvalue. In the second part of the paper, we consider the case when the
function has precisely one local minimum and one saddle point. We also
discuss further examples of supersymmetric operators, including the Witten
Laplacian and the infinitesimal generator for the time evolution of a chain of
classical anharmonic oscillators
Semiclassical analysis for the Kramers-Fokker-Planck equation
We study some accurate semiclassical resolvent estimates for operators that
are neither selfadjoint nor elliptic, and applications to the Cauchy problem.
In particular we get a precise description of the spectrum near the imaginary
axis and precise resolvent estimates inside the pseudo-spectrum. We apply our
results to the Kramers-Fokker-Planck operator