Kinetic and Related Models
April 2018 , Volume 11 , Issue 2
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Mark Kac introduced what is now called 'the Kac Walk' with the aim of investigating the spatially homogeneous Boltzmann equation by probabilistic means. Much recent work, discussed below, on Kac's program has run in the other direction: using recent results on the Boltzmann equation, or its one-dimensional analog, the non-linear Kac-Boltzmann equation, to prove results for the Kac Walk. Here we investigate new functional inequalities for the Kac Walk pertaining to entropy production, and introduce a new form of 'chaoticity'. We then show how these entropy production inequalities imply entropy production inequalities for the Kac-Boltzmann equation. This results validate Kac's program for proving results on the non-linear Boltzmann equation via analysis of the Kac Walk, and they constitute a partial solution to the 'Almost' Cercignani Conjecture on the sphere.
We consider a one dimensional infinite chain of harmonic oscillators whose dynamics is weakly perturbed by a stochastic term conserving energy and momentum and whose evolution is governed by an Ornstein-Uhlenbeck process. We prove the kinetic limit for the Wigner functions corresponding to the chain. This result generalizes the results of [
In this paper, we investigate the use of so called "duality lemmas" to study the system of discrete coagulation-fragmentation equations with diffusion. When the fragmentation is strong enough with respect to the coagulation, we show that we have creation and propagation of superlinear moments. In particular this implies that strong enough fragmentation can prevent gelation even for superlinear coagulation, a statement which was only known up to now in the homogeneous setting. We also use this control of superlinear moments to extend a recent result from [
We study the time evolution of the three dimensional Vlasov-Poisson plasma interacting with a positive point charge in the case of infinite mass. We prove the existence and uniqueness of the classical solution to the system by assuming that the initial density slightly decays in space, but not integrable. This result extends a previous theorem for Yukawa potential obtained in [
We study here a Fokker-Planck equation with variable coefficient of diffusion and boundary conditions which appears in the study of the wealth distribution in a multi-agent society [
We study a kinetic Vlasov/Fokker-Planck equation perturbed by a stochastic forcing term. When the noise intensity is not too large, we solve the corresponding Cauchy problem in a space of functions ensuring good localization in the velocity variable. Then we show under similar conditions that the generated dynamics, with prescribed total mass, admits a unique invariant measure which is exponentially mixing. The proof relies on hypocoercive estimates and hypoelliptic regularity. At last we provide an explicit example showing that our analytic framework does require some smallness condition on the noise intensity.
We present several regularity results for a biological network formulation model originally introduced by D. Cai and D. Hu [
In this work, we propose numerical schemes for linear kinetic equation, which are able to deal with a diffusion limit and an anomalous time scale of the form
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