Mixtures are convex combinations of laws. Despite this simple definition, a mixture can be far more subtle than its mixed components. For instance, mixing gaussian laws may produce a potential with multiple deep wells. We study in the present work fine properties of mixtures with respect to concentration of measure and Sobolev type functional inequalities. We provide sharp Laplace bounds for Lipschitz functions in the case of generic mixtures, involving a transportation cost diameter of the mixed...
We investigate the dissipativity properties of a class of scalar second
order parabolic partial differential equations with time-dependent
coefficients. We provide explicit condition on the drift term which ensure
that the relative entropy of one particular orbit with respect to some other
one decreases to zero. The decay rate is obtained explicitly by the use of
a Sobolev logarithmic inequality for the associated semigroup, which is
derived by an adaptation of Bakry's -calculus.
As a byproduct,...
We consider a Vlasov-Fokker-Planck equation governing the evolution
of the density of interacting and diffusive matter in the space of
positions and velocities.
We use a probabilistic interpretation to obtain convergence towards
equilibrium in Wasserstein distance with an explicit exponential
rate. We also prove a propagation of chaos property for an
associated particle system, and give rates on the approximation of
the solution by the particle system. Finally, a transportation
inequality...
Download Results (CSV)