Related papers: Existence and Exponential mixing of infinite white…
We proved the existence of an infinite dimensional stochastic system driven by white $\alpha$-stable noises ($1<\alpha \leq 2$), and prove this system is strongly mixing. Our method is by perturbing Ornstein-Uhlenbeck $\alpha$-stable…
Analysis of an interface stabilised finite element method for the scalar advection-diffusion-reaction equation is presented. The method inherits attractive properties of both continuous and discontinuous Galerkin methods, namely the same…
We provide deterministic controllability conditions that imply exponential mixing properties for randomly forced constrained dynamical systems with possibly unbounded state space. As an application, new ergodicity results are obtained for…
We show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in infinite dimensions. We concentrate on semilinear systems driven by cylindrical $\alpha$-stable noises,…
We develop and implement new probabilistic strategy for proving basic results about long time behaviour for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process as…
We consider a damped linear hyperbolic system modelling the propagation of pressure waves in a network of pipes. Well-posedness is established via semi-group theory and the existence of a unique steady state is proven in the absence of…
Integro-differential equations, analyzed in this work, comprise an important class of models of continuum media with nonlocal interactions. Examples include peridynamics, population and opinion dynamics, the spread of disease models, and…
In this paper, we prove that weak solutions of 3D stochastic primitive equations have exponential mixing property if the noise is sufficiently smooth and non-degenerate. With the help of uniqueness of strong solution of 3D stochastic…
We consider the numerical approximation of linear damped wave systems by Galerkin approximations in space and appropriate time-stepping schemes. Based on a dissipation estimate for a modified energy, we prove exponential decay of the…
We consider interacting particle systems with unbounded interaction range on general countably infinite graphs $S$ and prove explicit non-asymptotic error bounds for approximations of the infinite-volume dynamics by systems of finitely many…
This paper studies the 1D stochastic Allen--Cahn equation on a bounded domain driven by localized white noise. We prove that the associated Markov process admits a unique invariant measure and is exponential mixing. The main challenge lies…
A fully discrete Galerkin scheme for a thermodynamically consistent transient Max-well-Stefan system for the mass particle densities, coupled to the Poisson equation for the electric potential, is investigated. The system models the…
We consider an infinite system of spiking neurons with a drift and both excitatory and inhibitory connections. We study conditions for non-explosiveness and the uniqueness of the invariant measure. In particular, we examine conditions that…
We propose a locally conservative enriched Galerkin scheme that preserves the physical bounds for an elliptic problem. To this end, we use a substantial over-penalization of the discrete solution's jumps to obtain optimal convergence. To…
This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a general class of spatially homogeneous kinetic equations in space dimensions $d \geq 3$. Our results cover in particular…
Certain Markov processes, or deterministic evolution equations, have the property that they are dual to a stochastic process that exhibits extinction versus unbounded growth, i.e., the total mass in such a process either becomes zero, or…
Bounded variation estimates of Galerkin approximations are established in order to extract an almost everywhere convergent subsequence of Galerkin approximations. As a result we prove existence of weak solutions of initial boundary value…
We consider a class of mechanical particle systems with deterministic particle-disk interactions coupled to Gibbs heat reservoirs at possibly different temperatures. We show that there exists a unique (non-equilibrium) steady state. This…
We establish a new criterion for exponential mixing of random dynamical systems. Our criterion is applicable to a wide range of systems, including in particular dispersive equations. Its verification is in nature related to several topics,…
We prove the exponential convergence to a unique invariant measure for locally damped nonlinear Schr\"odinger equations, perturbed by bounded noise acting on only two Fourier modes. To tackle the lack of smoothing effect, we introduce…