1,887 research outputs found
From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules
We prove a quantitative result of convergence of a conservative stochastic
particle system to the solution of the homogeneous Landau equation for hard
potentials. There are two main difficulties: (i) the known stability results
for this class of Landau equations concern regular solutions and seem difficult
to extend to study the rate of convergence of some empirical measures; (ii) the
conservativeness of the particle system is an obstacle for (approximate)
independence. To overcome (i), we prove a new stability result for the Landau
equation for hard potentials concerning very general measure solutions. Due to
(ii), we have to couple, our particle system with some non independent
nonlinear processes, of which the law solves, in some sense, the Landau
equation. We then prove that these nonlinear processes are not so far from
being independent. Using finally some ideas of Rousset [25], we show that in
the case of Maxwell molecules, the convergence of the particle system is
uniform in time
Smoothness of the law of some one-dimensional jumping S.D.E.s with non-constant rate of jump
We consider a one-dimensional jumping Markov process ,
solving a Poisson-driven stochastic differential equation. We prove that the
law of admits a smooth density for , under some regularity and
non-degeneracy assumptions on the coefficients of the S.D.E. To our knowledge,
our result is the first one including the important case of a non-constant rate
of jump. The main difficulty is that in such a case, the map
is not smooth. This seems to make impossible the use of Malliavin calculus
techniques. To overcome this problem, we introduce a new method, in which the
propagation of the smoothness of the density is obtained by analytic arguments
A new regularization possibility for the Boltzmann equation with soft potentials
We consider a simplified Boltzmann equation: spatially homogeneous,
two-dimensional, radially symmetric, with Grad's angular cutoff, and linearized
around its initial condition. We prove that for a sufficiently singular
velocity cross section, the solution may become instantaneously a function,
even if the initial condition is a singular measure. To our knowledge, this is
the first regularization result in the case with cutoff: all the previous
results were relying on the non-integrability of the angular cross section.
Furthermore, our result is quite surprising: the regularization occurs for
initial conditions that are not too singular, but also not too regular. The
objective of the present work is to explain that the singularity of the
velocity cross section, which is often considered as a (technical) obstacle to
regularization, seems on the contrary to help the regularization
Finiteness of entropy for the homogeneous Boltzmann equation with measure initial condition
We consider the spatially homogeneous Boltzmann equation for (true) hard
and moderately soft potentials. We assume that the initial condition is a
probability measure with finite energy and is not a Dirac mass. For hard
potentials, we prove that any reasonable weak solution immediately belongs to
some Besov space. For moderately soft potentials, we assume additionally that
the initial condition has a moment of sufficiently high order ( is enough)
and prove the existence of a solution that immediately belongs to some Besov
space. The considered solutions thus instantaneously become functions with a
finite entropy. We also prove that in any case, any weak solution is
immediately supported by .Comment: Published in at http://dx.doi.org/10.1214/14-AAP1012 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Well-posedness of the spatially homogeneous Landau equation for soft potentials
We consider the spatially homogeneous Landau equation of kinetic theory, and
provide a differential inequality for the Wasserstein distance with quadratic
cost between two solutions. We deduce some well-posedness results. The main
difficulty is that this equation presents a singularity for small relative
velocities. Our uniqueness result is the first one in the important case of
soft potentials. Furthermore, it is almost optimal for a class of moderately
soft potentials, that is for a moderate singularity. Indeed, in such a case,
our result applies for initial conditions with finite mass, energy, and
entropy. For the other moderatley soft potentials, we assume additionnally some
moment conditions on the initial data. For very soft potentials, we obtain only
a local (in time) well-posedness result, under some integrability conditions.
Our proof is probabilistic, and uses a stochastic version of the Landau
equation, in the spirit of Tanaka
Stability of the stochastic heat equation in
We consider the white-noise driven stochastic heat equation on
with Lipschitz-continuous drift and diffusion
coefficients and . We derive an inequality for the
-norm of the difference between two solutions. Using some
martingale arguments, we show that this inequality provides some {\it a priori}
estimates on solutions. This allows us to prove the strong existence and
(partial) uniqueness of weak solutions when the initial condition belongs only
to , and the stability of the solution with respect to this initial
condition. We also obtain, under some conditions, some results concerning the
large time behavior of solutions: uniqueness of the possible invariant
distribution and asymptotic confluence of solutions
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