Existence Of Invariant Measure Research Articles

Existence of Invariant Measures for Reflected Stochastic Partial Differential Equations

In this article, we close a gap in the literature by proving existence of invariant measures for reflected stochastic partial differential equations with only one reflecting barrier. This is done by arguing that the sequence (u(t,cdot ))_{t ge 0} is tight in the space of probability measures on continuous functions and invoking the Krylov–Bogolyubov theorem. As we no longer have an a priori bound on our solution as in the two-barrier case, a key aspect of the proof is the derivation of a suitable L^p bound which is uniform in time.

Dynamics of stochastic reaction–diffusion lattice systems driven by nonlinear noise

This paper is concerned with the dynamics of the stochastic reaction–diffusion lattice systems defined on the entire integer set driven by locally Lipschitz nonlinear noise. We prove the existence and uniqueness of solutions as well as weak pullback random attractors for the mean random dynamical systems generated by the solution operators. The existence of invariant measures in ℓ2 for the stochastic equations is also established. The idea of uniform estimates on the tails of the solutions is employed to show the tightness of a family of probability distributions of the solutions.

On invariant measures and the asymptotic behavior of a stochastic delayed SIRS epidemic model

In this paper, we consider a stochastic epidemic model with time delay and general incidence rate. We first prove the existence and uniqueness of the global positive solution. By using the Krylov–Bogoliubov method, we obtain the existence of invariant measures. Furthermore, we study a special case where the incidence rate is bilinear with distributed time delay. When the basic reproduction number R0<1, the analysis of the asymptotic behavior around the disease-free equilibrium E0 is provided while when R0>1, we prove that the invariant measure is unique and ergodic. The numerical simulations also validate our analytical results.

Well-posedness of the stochastic Boussinesq equation driven by Levy processes

In this paper, we develop a new progressive stopping time technique to prove the existence and uniqueness of a special type of global solutions for the stochastic Boussinesq equations driven by Levy processes. Then we prove the existence of invariant measure.

Stochastic generalized magnetohydrodynamics equations with not regular multiplicative noise: Well-posedness and invariant measure

In this paper we study a stochastic magnetohydrodynamics (MHD) system with fractional diffusion and resistivity ( − Δ ) α , α > 0 , in R d , d = 2 , 3 . Our main goal is to identify the conditions on α under which we can prove the existence of a martingale solution, the pathwise uniqueness of solution and the existence of invariant measure when the noises are multiplicative and take values in functional space bigger than the space of square integrable functions. Roughly speaking, we prove that if α ≥ 1 , θ ∈ ( 0 , α ) and the driving noises take values in H − θ , then the stochastic system has at least a weak martingale solution. We also establish the pathwise uniqueness of solution whenever α ≥ d 2 . Finally, under the latter condition and under the addition of linear damping to the equations we are able to establish the existence of an invariant measure.

Isotropic Realizability of Fields and Reconstruction of Invariant Measures under Positivity Properties. Asymptotics of the Flow by a Non-Ergodic Approach

The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function σ such that σb is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure σ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b · v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.

Ergodicity and Kolmogorov Equations for Dissipative SPDEs with Singular Drift: a Variational Approach

We prove existence of invariant measures for the Markovian semigroup generated by the solution to a parabolic semilinear stochastic PDE whose nonlinear drift term satisfies only a kind of symmetry condition on its behavior at infinity, but no restriction on its growth rate is imposed. Thanks to strong integrability properties of invariant measures μ, solvability of the associated Kolmogorov equation in L1(μ) is then established, and the infinitesimal generator of the transition semigroup is identified as the closure of the Kolmogorov operator. A key role is played by a generalized variational setting.

Some criteria for the existence of invariant measures and asymptotic stability for random dynamical systems on polish spaces

ABSTRACTTomasz Szarek presented interesting criteria for the existence of invariant measures and asymptotic stability of Markov operators on Polish spaces. Hans Crauel in his book presented the theory of random probabilistic measures on Polish spaces showing that notions of compactness and tightness for such measures are in one-to-one correspondence with such notions for non-random measures on Polish spaces, in addition to the criteria under which the space of random measures is itself a Polish space. This result allowed the transfer of results of Szarek to the case of random dynamical systems in the sense of Arnold. These criteria are interesting because they allow to use the existence of simple deterministic Lyapunov type function together with additional conditions to show the existence of invariant measures and asymptotic stability of random dynamical systems on general Polish spaces.

Existence of invariant measures for the stochastic damped KdV equation

We address the long time behavior of solutions of the stochastic Korteweg-de Vries equation $ du + (\partial^3_x u +u\partial_x u +\lambda u)dt = f dt+\Phi dW_t$ on ${\mathbb R}$ where $f$ is a deterministic force. We prove that the Feller property holds and establish the existence of an invariant measure. The tightness is established with the help of the asymptotic compactness, which is carried out using the Aldous criterion.

Invariant measures on multi-valued functions

In this paper we consider the question of under which conditions multi-valued dynamical systems admit invariant measures. We give results on the existence of invariant measures with full support on orbit spaces of multi-valued dynamical systems. We use these measures on the orbit space to induce measures on the original dynamical system. We focus on the question of when a non-atomic invariant measure on the orbit space induces an atomic invariant measure on the multi-valued dynamical system. This phenomenon is an indicator of complicated multi-periodic behaviour.

Existence of invariant measures for some damped stochastic dispersive equations

We address the long-time behavior of solutions to damped dispersive stochastic partial differential equations, namely the KdV equation and the nonlinear Schrödinger equation on the whole space. We prove that the transition semigroup is Feller and establish the existence of an invariant measure using the asymptotic compactness property of the transition semigroup and the Aldous criterion.

Existence of invariant measures for the stochastic damped Schrödinger equation

In this paper, we address the long time behavior of solutions of the stochastic Schrodinger equation $$du+(\lambda u+i\Delta u +i\alpha |u|^{2\sigma }u)dt=\Phi dW_t$$ in $${{\mathbb {R}}}^{d}$$ . We prove the existence of an invariant measure in $$H^{1}$$ for $$\sigma <2/(2-d)$$ in the defocusing case and for $$\sigma <2/d$$ in the focusing case. We also establish asymptotic compactness of invariant measures, implying in particular the existence of an ergodic measure.

On Unique Ergodicity in Nonlinear Stochastic Partial Differential Equations

We illustrate how the notion of asymptotic coupling provides a flexible and intuitive framework for proving the uniqueness of invariant measures for a variety of stochastic partial differential equations whose deterministic counterpart possesses a finite number of determining modes. Examples exhibiting parabolic and hyperbolic structure are studied in detail. In the later situation we also present a simple framework for establishing the existence of invariant measures when the usual approach relying on the Krylov-Bogolyubov procedure and compactness fails.

On the existence of an invariant measure for isotropic diffusions in random environment

The results of this paper build upon those first obtained by Sznitman and Zeitouni (Invent Math 164(3), 455–567, 2006). We establish, for spacial dimensions dge 3, the existence of a unique invariant measure for isotropic diffusions in random environment on mathbb {R}^d which are small perturbations of Brownian motion. Furthermore, we establish a general homogenization result for initial data which are locally measurable with respect to the coefficients.

Properties of stochastic integro-differential equations with infinite delay: Regularity, ergodicity, weak sense Fokker–Planck equations

This work focuses on properties of stochastic integro-differential equations with infinite delay (or unbounded delay). Our main approach is to map the solution processes into another Polish space. Under suitable conditions, it is shown that the resulting processes are Markov. Furthermore, sufficient conditions for Feller properties, recurrence, ergodicity, and existence of invariant measures are obtained. Moreover, weak sense Fokker–Planck equations are derived for the underlying processes.

Invariant measures of smooth dynamical systems, generalized functions and summation methods

We discuss conditions for the existence of invariant measures of smooth dynamical systems on compact manifolds. If there is an invariant measure with continuously differentiable density, then the divergence of the vector field along every solution tends to zero in the Cesàro sense as time increases unboundedly. Here the Cesàro convergence may be replaced, for example, by any Riesz summation method, which can be arbitrarily close to ordinary convergence (but does not coincide with it). We give an example of a system whose divergence tends to zero in the ordinary sense but none of its invariant measures is absolutely continuous with respect to the `standard' Lebesgue measure (generated by some Riemannian metric) on the phase space. We give examples of analytic systems of differential equations on analytic phase spaces admitting invariant measures of any prescribed smoothness (including a measure with integrable density), but having no invariant measures with positive continuous densities. We give a new proof of the classical Bogolyubov–Krylov theorem using generalized functions and the Hahn–Banach theorem. The properties of signed invariant measures are also discussed.

On metric properties of unconventional limit sets of contractive non-Archimedean dynamical systems

ABSTRACTIn this paper, we define the limit set Λξ of an unconventional set of contractive functions {fk} on the unit ball of non-Archimedean algebra. Then, we prove that Λξ is compact, perfect and uniformly disconnected. It is shown that there is a new collection of contractive mappings defined on Λξ. Moreover, we establish that the set Λξ coincides with the limit set generated by the semi-group of . This result allows us to further investigate the structure of Λξ by means of this limiting set. As an application, we demonstrate the existence of invariant measures on Λξ. We should stress that the non-Archimedeanity of the space is essentially used in the paper. Therefore, the methods applied in this paper are not longer valid in the Archimedean setting (i.e. in case of real or complex numbers).

Ergodicity for a stochastic Hodgkin–Huxley model driven by Ornstein–Uhlenbeck type input

Nous considerons un modele stochastique decrivant un neurone stimule par un signal qu’il recoit a travers son systeme dendritique. Ce signal aleatoire, de type Ornstein–Uhlenbeck, resulte de la perturbation d’un signal periodique deterministe. Pour decrire l’evolution du potentiel de membrane du neurone, nous faisons appel au modele classique de Hodgkin et Huxley. Le systeme stochastique obtenu est fortement degenere. Nous demontrons que la condition de Hormander faible est satisfaite en tout point de l’espace d’etats. L’existence d’une fonction de Lyapunov permet de conclure qu’il existe au plus un nombre fini de mesures ergodiques pour ce modele. Par consequent, la complexite du systeme est reduite de maniere significative par rapport a la situation deterministe.

Stability in distribution of neutral stochastic functional differential equations

In this note, by the weak convergence approach, we investigate stability in distribution for a range of neutral stochastic functional differential equations. As a byproduct, our result derived also implies existence of invariant measure of the semigroup generated by the segment process. Last but not least, we close some gaps of Lemma 2, and improve greatly Lemma 4 due to Hu and Wang (2012).

Atomic operators, random dynamical systems and invariant measures

It is proved that the existence of invariant measures for families of the so-called atomic operators (nonlinear generalized weighted shifts) defined over spaces of measurable functions follows from the existence of appropriate invariant bounded sets. Typically, such operators come from infinite-dimensional stochastic differential equations generating not necessarily regular solution flows, for instance, from stochastic differential equations with time delay in the diffusion term (regular solution flows called also Carathéodory flows are those almost surely continuous with respect to the initial data). Thus, it is proved that to ensure the existence of an invariant measure for a stochastic solution flow it suffices to find a bounded invariant subset, and no regularity requirement for the flow is necessary. This result is based on the possibility to extend atomic operators by continuity to a suitable set of Young measures, which is proved in the paper. A motivating example giving a new result on the existence of an invariant measure for a possibly nonregular solution flow of some model stochastic differential equation is also provided.