Related papers: Invariant measures for the open KPZ equation: an a…
The ergodic hypothesis is examined for energetically open fluid systems represented by the barotropic Navier--Stokes equations with general inflow/outflow boundary conditions. We show that any globally bounded trajectory generates a…
We consider invariant measures for the stochastic Burgers equation on $\mathbb{R}$, forced by the derivative of a spacetime-homogeneous Gaussian noise that is white in time and smooth in space. An invariant measure is indecomposable, or…
The paper is concerned with a nonlinear system of partial differential equations with parameters and the random external force. This system describes the two-layer quasi-solenoidal Lorenz model for a baroclinic atmosphere on a rotating…
Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a…
Quantum ergodicity asserts that almost all infinite sequences of eigenstates of a quantized ergodic system are equidistributed in the phase space. On the other hand, there are might exist exceptional sequences which converge to different…
Improved rates of convergence for ergodic homogeneous Markov chains are studied. In comparison to the earlier papers the setting is also generalised to the case without a unique dominated measure. Examples are provided where the new bound…
In this paper we deal with an invariant ergodic hyperbolic measure $\mu$ for a diffeomorphism $f,$ assuming that $f$ it is either $C^{1+\alpha}$ or $f$ is $C^1$ and the Oseledec splitting of $\mu$ is dominated. We show that this system…
For a Markov chain $Y$ with values in a Polish space, consider the entrance chain, obtained by sampling $Y$ at the moments when it enters a fixed set $A$ from its complement $A^c$. Similarly, consider the exit chain, obtained by sampling…
In this paper we provide sufficient conditions which guarantee the existence of a system of invariant measures for semigroups associated to systems of parabolic differential equations with unbounded coefficients. We prove that these…
For an ergodic hyperbolic measure $\omega$ of a $C^{1+{\alpha}}$ diffeomorphism, there is an $\omega$ full-measured set $\tilde\Lambda$ such that every nonempty, compact and connected subset $V$ of $\mathbb{M}_{inv}(\tilde\Lambda)$…
We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by…
We prove that for certain partially hyperbolic skew-products, non-uniform hyperbolicity along the leaves implies existence of a finite number of ergodic absolutely continuous invariant probability measures which describe the asymptotics of…
In this paper, we study ergodic features of invariant measures for the partially hyperbolic horseshoe at the boundary of uniformly hyperbolic diffeomorphisms constructed in \cite{DHRS07}. Despite the fact that the non-wandering set is a…
We establish convergence to an invariant measure as time tends to infinity, for a large class of (possibly non-Markovian) stochastic volatility models. Our arguments are based on a novel coupling idea for Markov chains which also extends to…
We prove that every Markov solution to the three dimensional Navier-Stokes equation with periodic boundary conditions driven by additive Gaussian noise is uniquely ergodic. The convergence to the (unique) invariant measure is exponentially…
Quantum trajectories are Markov processes that describe the time-evolution of a quantum system undergoing continuous indirect measurement. Mathematically, they are defined as solutions of the so-called "Stochastic Schr\"odinger Equations",…
We consider for the full time-dependent Maxwell's equations the inverse problem of identifying locations and certain properties of small electromagnetic inhomogeneities in a homogeneous background medium from dynamic boundary measurements…
This paper studies the large scale limits of multi-type invariant distributions and Busemann functions of planar stochastic growth models in the Kardar-Parisi-Zhang (KPZ) class. We identify a set of sufficient hypotheses for convergence of…
We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in…
We consider dynamics of scalar semilinear parabolic equations on bounded intervals with periodic boundary conditions, and on the entire real line, with a general nonlinearity $g(t,x,u,u_x)$ either not depending on $t$, or periodic in $t$.…