Random Evolution Equations Research Articles

Random attractors for a stochastic nonlocal delayed reaction–diffusion equation on a semi-infinite interval

Abstract The aim of this paper is to prove the existence and qualitative property of random attractors for a stochastic non-local delayed reaction–diffusion equation (SNDRDE) on a semi-infinite interval with a Dirichlet boundary condition at the finite end. This equation models the spatial–temporal evolution of the mature individuals for a two-stage species whose juvenile and adults both diffuse that lives on a semi-infinite domain and subject to random perturbations. By transforming the SNDRDE into a random evolution equation with delay, by means of a stationary conjugate transformation, we first establish the global existence and uniqueness of solutions to the equation, after which we show the solutions generate a random dynamical system. Then, we deduce uniform a priori estimates of the solutions and show the existence of bounded random absorbing sets. Subsequently, we prove the pullback asymptotic compactness of the random dynamical system generated by the SNDRDE with respect to the compact open topology, and hence obtain the existence of random attractors. At last, it is proved that the random attractor is an exponentially attracting stationary solution under appropriate conditions. The theoretical results are illustrated by application to the stochastic non-local delayed Nicholson’s blowfly equation.

SUPPORT THEOREM FOR THE POSITIVE RANDOM EVOLUTION EQUATION IN H"{O}LDER NORM

Let's consider the stochastic differential equation \[\left\{ \begin{split} dX_t&=\sigma(X_t,V_t )dW_t+b(X_t,R_t)dt\\ X_0&=x>0 \end{split}\right. \] In this paper, we establish the support theorem for this positive random evolution equation type on $C^{\alpha,0}([0;1];\mathbb{R})$. We use the linear interpolations of $W$ for the proof.

Random Evolution Equations: Well-Posedness, Asymptotics, and Applications to Graphs

We study diffusion-type equations supported on structures that are randomly varying in time. After settling the issue of well-posedness, we focus on the asymptotic behavior of solutions: our main result gives sufficient conditions for pathwise convergence in norm of the (random) propagator towards a (deterministic) steady state. We apply our findings in two environments with randomly evolving features: ensembles of difference operators on combinatorial graphs, or else of differential operators on metric graphs.

Stroock-Varadhan support theorem for random evolution equation in Besov-Orlicz spaces

We consider the family of stochastic processes $X=\{X_t, t\in [0;1]\}\,,$ where $X$ is the solution of the It\^{o} stochastic differential equation \[dX_t = \sigma(X_t, Z_t)dW_t + b(X_t,Y_t) dt \hspace*{2cm}\] whose coefficients Lipschitzian depend on $Z=\{Z_t, t\in [0;1]\} $ and $Y=\{Y_t, t\in [0;1]\}$. We prove that the trajectories of $X$ a.s. belong to the Besov-Orlicz space defined by the f nction $M(x)=e^{x^2}-1$ and the modulus of continuity $\omega(t)=\sqrt{t\log(1/t)}$. The aim of this work is to characterize the support of the law $X$ in this space.

Random time change and related evolution equations. Time asymptotic behavior

In this paper, we investigate the time asymptotic behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The classes of subordinators for which asymptotic analysis may be realized are described.

Existence of solutions for quasilinear random impulsive neutral differential evolution equation

This paper deals with the existence of solutions for quasilinear random impulsive neutral functional differential evolution equation in Banach spaces and the results are derived by using the analytic semigroup theory, fractional powers of operators and the Schauder fixed point approach. An application is provided to illustrate the theory.

Stability of transition waves and positive entire solutions of Fisher-KPP equations with time and space dependence

This paper is concerned with the stability of transition waves and strictly positive entire solutions of random and nonlocal dispersal evolution equations of Fisher-KPP type with general time and space dependence, including time and space periodic or almost periodic dependence as special cases. We first show the existence, uniqueness, and stability of strictly positive entire solutions of such equations. Next, we show the stability of uniformly continuous transition waves connecting the unique strictly positive entire solution and the trivial solution zero and satisfying certain decay property at the end close to the trivial solution zero (if it exists). The existence of transition waves has been studied in Liang and Zhao (2010 J. Funct. Anal. 259 857–903), Nadin (2009 J. Math. Pures Appl. 92 232–62), Nolen et al (2005 Dyn. PDE 2 1–24), Nolen and Xin (2005 Discrete Contin. Dyn. Syst. 13 1217–34) and Weinberger (2002 J. Math. Biol. 45 511–48) for random dispersal Fisher-KPP equations with time and space periodic dependence, in Nadin and Rossi (2012 J. Math. Pures Appl. 98 633–53), Nadin and Rossi (2015 Anal. PDE 8 1351–77), Nadin and Rossi (2017 Arch. Ration. Mech. Anal. 223 1239–67), Shen (2010 Trans. Am. Math. Soc. 362 5125–68), Shen (2011 J. Dynam. Differ. Equ. 23 1–44), Shen (2011 J. Appl. Anal. Comput. 1 69–93), Tao et al (2014 Nonlinearity 27 2409–16) and Zlatoš (2012 J. Math. Pures Appl. 98 89–102) for random dispersal Fisher-KPP equations with quite general time and/or space dependence, and in Coville et al (2013 Ann. Inst. Henri Poincare 30 179–223), Rawal et al (2015 Discrete Contin. Dyn. Syst. 35 1609–40) and Shen and Zhang (2012 Comm. Appl. Nonlinear Anal. 19 73–101) for nonlocal dispersal Fisher-KPP equations with time and/or space periodic dependence. The stability result established in this paper implies that the transition waves obtained in many of the above mentioned papers are asymptotically stable for well-fitted perturbation. Up to the author’s knowledge, it is the first time that the stability of transition waves of Fisher-KPP equations with general time and space dependence is studied.

Support Theorem for Random Evolution Equations in Hölderian Norm

Support Theorem for Random Evolution Equations in Hölderian Norm

Attractors for a random evolution equation with infinite memory: Theoretical results

The long-time behavior of solutions (more precisely, the existence of random pullback attractors) for an integro-differential parabolic equation of diffusion type with memory terms, more particularly with terms containing both finite and infinite delays, as well as some kind of randomness, is analyzed in this paper. We impose general assumptions not ensuring uniqueness of solutions, which implies that the theory of multivalued dynamical system has to be used. Furthermore, the emphasis is put on the existence of random pullback attractors by exploiting the techniques of the theory of multivalued nonautonomous/random dynamical systems.

Random attractor for stochastic non-autonomous damped wave equation with critical exponent

In this paper, we prove the existence of random attractor and obtainan upper bound of fractal dimension of random attractor forstochastic non-autonomous damped wave equation with criticalexponent and additive white noise. We first prove the existence of arandom attractor by carefully splitting the positivity of the linearoperator in the corresponding random evolution equation of the firstorder in time and by carefully decomposing the solutions of systemthrough two different modes, and we show the boundedness of randomattractor in a higher regular space by a recurrence method. Then weestablish a criterion to bound the fractal dimension of a randominvariant set for a cocycle and applied these conditions to get anupper bound of fractal dimension of the random attractor ofconsidered system.

A Class of Impulsive Stochastic Parabolic Functional Differential Equations and Their Asymptotics

This paper is devoted to the study of the initial value problem for a class of semilinear impulsive stochastic parabolic functional differential equations. Incorporating certain positive operators, these equations have as archetype impulsive stochastic functional heat differential equations supplemented by homogeneous Dirichlet boundary conditions. By the classical Galerkin's method, we prove a well-posedness result for the initial value problem for a class of linear random evolution equation which is necessary in developing our main theory. Using this well-posedness result and the classical contraction mapping argument, we prove that the initial value problem under consideration is globally well-posed. Employing the technique of Razumikhin, we prove under some additional assumptions that the trivial solution of the equation in question is mean-square exponentially stable.

Random attractors for stochastic lattice dynamical systems with infinite multiplicative white noise

In this paper we investigate the long term behavior of a stochastic lattice dynamical system with a diffusive nearest neighbor interaction, a dissipative nonlinear reaction term, and a different multiplicative white noise at each node. We prove that this stochastic lattice equation generates a random dynamical system that possesses a global random attractor. In particular, we first establish an existence theorem for weak solutions to general random evolution equations, which is later applied to the specific stochastic lattice system to show that it has weak solutions and the solutions generate a random dynamical system. We then prove the existence of a random attractor of the underlying random dynamical system by constructing a random compact absorbing set and using an embedding theorem. The major novelty of this work is that we consider a different multiplicative white noise term at each different node, which significantly improves the previous results in the literature where the same multiplicative noise was considered at all the nodes. As a consequence, the techniques used in the existing literature are not applicable here and a new methodology has to be developed to study such systems.

Averaging of Attractors and Inertial Manifolds for Parabolic PDE With Random Coefficients

Abstract The averaging method has been used to study random or non-autonomous systems on a fast time scale. We apply this method to a random abstract evolution equation on a fast time scale whose long time behavior can be characterized by a random attractor or a random inertial manifold. The main purpose is to show that the long-time behavior of such a system can be described by a deterministic evolution equation with averaged coefficients. Our first result provides an averaging result on finite time intervals which we use to show that under a dissipativity assumption the attractors of the fast time scale systems are upper semicontinuous when the scaling parameter goes to zero. Our main result deals with a global averaging procedure. Under some spectral gap condition we show that inertial manifolds of the fast time scale system tend to an inertial manifold of the averaged system when the scaling parameter goes to zero. These general results can be applied to semilinear parabolic differential equations containing a scaled ergodic noise on a fast time scale which includes scaled almost periodic motions.

Stability for a random evolution equation with Gaussian perturbation

In this paper we consider a random evolution equation which is perturbed by Gaussian type noise, and show how to construct its coupled solutions. On the basis of the coupling results, we discuss the asymptotic flatness and the stability in total variation norm for the solutions of the equation. In addition, we also prove the Feller continuity, and the existence and uniqueness of invariant measures of the solutions.

Invariant Measures for a Random Evolution Equation with Small Perturbations

Invariant Measures for a Random Evolution Equation with Small Perturbations

Invariant Measures for a Random Evolution Equation with Small Perturbations

In this paper we consider a random evolution equation with small perturbations, and show how to construct coupled solutions to the equations. As applications, we prove the Feller continuity of the solutions and the existence and uniqueness of invariant measures. Furthermore, we establish a large deviations principle for the family of invariant measures as the perturbations tend to zero.

A large deviations principle for small perturbations of random evolution equation in Hölder norm

A large deviations principle for small perturbations of random evolution equation in Hölder norm

A unified approach to the large deviations for small perturbations of random evolution equations

LetXɛ = {Xɛ (t ; 0 ⩽t ⩽ 1 } (ɛ > 0) be the processes governed by the following stochastic differential equations: $$dX^\varepsilon (t) = \sqrt \varepsilon \sigma (X^\varepsilon (t))dB(t) + b(X^\varepsilon (t),\nu (t))dt,$$ wherev(t) is a random process independent of the Brownian motionB(·). Some large deviation (LD) properties of { (Xɛ, ν(.)); ɛ > 0} are proved. For a particular case, an explicit representation of the rate function is also given, which solves a problem posed by Eizenberg and Freidlin. In the meantime, an abstract LD theorem is obtained.

A large deviation principle for small perturbations of random evolution equations in Hölder norm

In this paper, it is shown that the Wentzell-Freidlin large deviations estimates for random evolution equations still hold with the sup-norm replaced by any Hölder norm with exponent α < 1 2 .

Averaging principle for perturbed random evolution equations and corresponding Dirichlet problems

We continue the study of exit problems for perturbed random evolution equations corresponding to the weakly coupled elliptic PDE systems. In the present paper we consider the cases where the corresponding random evolutions stay in a given domain for ever with probability one, but do not hinder the exit of the perturbed process. We treat such problems by methods based on the averaging principle. In such a way we also study the asymptotic behavior of the solutions of the corresponding perturbed Dirichlet problems.