arXiv subjects

B. Rüdiger

Publications and source records attributed to B. Rüdiger.

3 recordsLinked to original sources

arXiv2024

Identification and existence of Boltzmann processes

The stochastic differential equation of McKean-Vlasov type is identified such that the Fokker-Planck equation associated to it is the Boltzmann equation. Hence, we call its solutions as Boltzmann processes. They describe the dynamics (in position and velocity) of particles expanding in vacuum in accordance with the Boltzmann equation. Given a solution $f:=$ $\{f(t,x,v\}_{0 \leq t \leq T} $ of the Boltzmann equation, the existence of solutions to the McKean-Vlasov SDE is established for the non-cutoff hard sphere case.

math.PR↗

arXiv2017

The Enskog Process

The existence of a weak solution to a McKean-Vlasov type stochastic differential system corresponding to the Enskog equation of the kinetic theory of gases is established under natural conditions. The distribution of any solution to the system at each fixed time is shown to be unique. The existence of a probability density for the time-marginals of the velocity is verified in the case where the initial condition is Gaussian, and is shown to be the density of an invariant measure.

math.PR↗

arXiv2016

Ito formula for mild solutions of SPDEs with Gaussian and non-Gaussian noise and applications to stability properties

We use Yosida approximation to find an Itô formula for mild solutions $\left\{X^x(t), t\geq 0\right\}$ of SPDEs with Gaussian and non-Gaussian coloured noise, the non Gaussian noise being defined through compensated Poisson random measure associated to a Lévy process. The functions to which we apply such Itô formula are in $C^{1,2}([0,T]\times H)$, as in the case considered for SDEs in [9]. Using this Itô formula we prove exponential stability and exponential ultimate boundedness properties in mean square sense for mild solutions. We also compare such Itô formula to an Itô formula for mild solutions introduced by Ichikawa in [8], and an Itô formula written in terms of the semigroup of the drift operator [11] which we extend before to the non Gaussian case.

math.PR↗