No abstract

In this work, we study ergodicity of continuous time Markov processes on state space R ≥0 := [0, ∞) obtained as unique strong solutions to stochastic equations with jumps. Our first main result establishes exponential ergodicity in the Wasserstein distance, provided the stochastic equation satisfies a comparison principle and the drift is dissipative. In particular, it is applicable to continuous-state branching processes with immigration (shorted as CBI processes), possibly with nonlinear branching mechanisms or in Lévy random environments. Our second main result establishes exponential ergodicity in total variation distance for subcritical CBI processes under a first moment condition on the jump measure for branching and a log-moment condition on the jump measure for immigration.

A dynamical version of the Widom-Rowlinsom model in the continuum is considered. The dynamics is modelled by a spatial two-component birth-and-death Glauber process where particles, in addition, are allowed to change their type with density dependent rates. An evolution of states is constructed as the unique weak solution to the associated Fokker-Planck equation. Such solution is obtained by means of its correlation functions which belong to a certain Ruelle space. Existence of a unique invariant measure and ergodicity with exponential rate is established. The mesoscopic limit is considered, it is related with the verification of the chaos preservation property.

We investigate the anisotropic stable JCIR process which is a multidimensional extension of the stable JCIR process but also a multi-dimensional analogue of the classical JCIR process. We prove that the heat kernel of the anisotropic stable JCIR process exists and it satisfies an a-priori bound in a weighted anisotropic Besov norm. Based on this regularity result we deduce the strong Feller property and prove, for the subcritical case, exponential ergodicity in total variation. Also, we show that in the one-dimensional case the corresponding heat kernel is smooth.

The density of a moderately dense gas evolving in a vacuum is given by the solution of an Enskog equation. Recently we have constructed in [ARS17] the stochastic process that corresponds to the Enskog equation under suitable conditions. The Enskog process is identified as the solution of a McKean-Vlasov equation driven by a Poisson random measure. In this work, we continue the study for a wider class of collision kernels that includes hard and soft potentials. Based on a suitable particle approximation of binary collisions, the existence of an Enskog process is established.Let us briefly comment on particular examples of collision kernels Bp|v´u|, nq in dimension d " 3. Boltzmann's original model was first formulated for (true) hard spheres where Bp|u´v|, nqdn " |pu´v, nq|dn.A transformation in polar coordinates to a system where the center is in u`v 2 and e 3 " p0, 0, 1q is parallel to u´v, i.e. e 3 |u´v| " u´v, leads to Bp|u´v|, nqdn " |pu´v, nq|dn " |u´v| sinˆθ 2˙c osˆθ 2˙d θdφ,where θ P p0, πs is the angle between u´v and u ‹´v‹ and φ P p0, 2πs is the longitude angle, see Tanaka [Tan79] or Horowitz and Karandikar [HK90]. More generally, many results rely on Grad's angular cut-off assumption where it is supposed that ż S d´1Bp|v´u|, nqdn ă 8,