We discuss various properties of Probabilistic Cellular Automata, such as the structure of the set of stationary measures and multiplicity of stationary measures (or phase transition) for reversible models.
The existence of steady states in the microcanonical case for a system describing the interaction of gravitationally attracting particles with a self-similar pressure term is proved. The system generalizes the Smoluchowski-Poisson equation. The presented theory covers the case of the model with diffusion that obeys the Fermi-Dirac statistic.
The existence of a one-parameter family of stationary solutions to a fragmentation equation with size diffusion is established. The proof combines a fixed point argument and compactness techniques.
We consider plasma tearing mode instabilities when the resistivity depends on a flux function (ψ), for the plane slab model. This problem, represented by the MHD equations, is studied as a bifurcation problem. For so doing, it is written in the form (I(.)-T(S,.)) = 0, where T(S,.) is a compact operator in a suitable space and S is the bifurcation parameter. In this work, the resistivity is not assumed to be a given quantity (as usually done in previous papers, see [1,2,5,7,8,9,10], but it depends...
Let be a three times partially differentiable function on , let be a collection of real-valued random variables and let be a multivariate Gaussian vector. In this article, we develop Stein’s method to give error bounds on the difference in cases where the coordinates of are not necessarily independent, focusing on the high dimensional case . In order to express the dependency structure we use Stein couplings, which allows for a broad range of applications, such as classic occupancy,...
Stimuli-responsive polymers result in on-demand regulation of properties and functioning of various nanoscale systems. In particular, they allow stimuli-responsive control of flow rates through membranes and nanofluidic devices with submicron channel sizes. They also allow regulation of drug release from nanoparticles and nanofibers in response to temperature or pH variation in the surrounding medium. In the present work two relevant mathematical models are introduced to address precipitation-driven...
This paper gives an approximation of the solution of the Boltzmann equation by stochastic interacting particle systems in a case of cut-off collision operator and small initial data. In this case, following the ideas of Mischler and Perthame, we prove the existence and uniqueness of the solution of this equation and also the existence and uniqueness of the solution of the associated nonlinear martingale problem. Then, we first delocalize the interaction by considering a mollified Boltzmann...
We introduce and investigate a set-valued analogue of classical Langevin equation on a Riemannian manifold that may arise as a description of some physical processes (e.g., the motion of the physical Brownian particle) on non-linear configuration space under discontinuous forces or forces with control. Several existence theorems are proved.
We show that recently introduced noncommutative -spaces can be used to constructions of Markov semigroups for quantum systems on a lattice.
We present a probabilistic model of the microscopic scenario of dielectric relaxation. We prove a limit theorem for random sums of a special type that appear in the model. By means of the theorem, we show that the presented approach to relaxation phenomena leads to the well known Havriliak-Negami empirical dielectric response provided the physical quantities in the relaxation scheme have heavy-tailed distributions. The mathematical model, presented here in the context of dielectric relaxation, can...