Nonlinear Multiplicative Noise Research Articles

Existence and Regularity of Random Attractors for Stochastic Evolution Equations Driven by Rough Noise

AbstractThis work establishes the existence and regularity of random pullback attractors for parabolic partial differential equations with rough nonlinear multiplicative noise under natural assumptions on the coefficients. To this aim, we combine tools from rough path theory and random dynamical systems. An application is given by partial differential equations with rough boundary noise, for which flow transformations are not available.

The Effects of Nonlinear Noise on the Fractional Schrödinger Equation

The aim of this work is to investigate the influence of nonlinear multiplicative noise on the Cauchy problem of the nonlinear fractional Schrödinger equation in the non-radial case. Local well-posedness follows from estimates related to the stochastic convolution and deterministic non-radial Strichartz estimates. Furthermore, the blow-up criterion is presented. Then, with the help of Itô’s lemma and stopping time arguments, the global solution is constructed almost surely. The main innovation is that the non-radial global solution is given under fractional-order derivatives and a nonlinear noise term.

Smoothness of invariant manifolds for stochastic evolution equations with non-dense domain

With the crucial estimation of random Stieltjes-type convolution, the Lyapunov–Perron method allows us to establish the existence of invariant dynamics for stochastic evolution equation with non-dense domain. In this paper, we develop the conditions for [Formula: see text]-smoothness of center manifold for such equation with linear multiplicative noise, unstable manifold and stable set for the one with nonlinear multiplicative noise as well. Our result may apply to a broad range of stochastic systems with non-dense domain beyond Brownian noise.

Existence of smooth stable manifolds for a class of parabolic SPDEs with fractional noise

Little seems to be known about the invariant manifolds for stochastic partial differential equations (SPDEs) driven by nonlinear multiplicative noise. Here we contribute to this aspect and analyze the Lu-Schmalfuß conjecture [Garrido-Atienza, et al., (2010) [14]] on the existence of stable manifolds for a class of parabolic SPDEs driven by nonlinear multiplicative fractional noise. We emphasize that stable manifolds for SPDEs are infinite-dimensional objects, and the classical Lyapunov-Perron method cannot be applied, since the Lyapunov-Perron operator does not give any information about the backward orbit. However, by means of interpolation theory, we construct a suitable function space in which the discretized Lyapunov-Perron-type operator has a unique fixed point. Based on this we further prove the existence and smoothness of local stable manifolds for such SPDEs.

The asymptotic behavior of solutions for stochastic evolution equations with pantograph delay

The polynomial stability problem of stochastic delay differential equations has been studied in recent years. In contrast, there are relatively few works on stochastic partial differential equations with pantograph delay. The present paper is devoted to investigating large-time asymptotic properties of solutions for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. We first show that the mild solutions of stochastic pantograph delay evolution equations with nonlinear multiplicative noise tend to zero with general decay rate (including both polynomial and logarithmic rates) in the [Formula: see text]th moment and almost sure senses. The analysis is based on the Banach fixed point theorem and various estimates involving the gamma function. Moreover, by using a generalized version of the factorization formula and exploiting an approximation technique and a convergence analysis, we construct the nontrivial equilibrium solution, defined for [Formula: see text], for stochastic pantograph delay evolution equations with nonlinear multiplicative noise. In particular, the uniqueness, Hölder regularity in time and general stability, in the [Formula: see text]th moment and almost sure senses, of the nontrivial equilibrium solution are established.

The Large Deviation of Semi-linear Stochastic Partial Differential Equation Driven by Brownian Sheet

We prove the the large deviation principle(LDP) for the law of the one-dimensional semilinear stochastic partial differential equations driven by nonlinear multiplicative noise. Firstly, combining the energy estimate and approximation procedure, we obtain the existence of global solution. Then the large deviation principle is obtained via weak convergence method.

Weak mean random attractors for nonautonomous stochastic parabolic equation with variable exponents

In this paper, we consider the asymptotic behavior of solutions for nonautonomous stochastic parabolic equation with nonstandard growth condition driven by nonlinear multiplicative noise for the first time. First, by making use of variational method, we prove the existence and uniqueness of solutions, and then the mean random dynamical systems generated by stochastic parabolic equations with variable exponents are obtained. Finally, due to the influence of variable indexes (dependent on space variable), we show the existence of weak mean random attractors under suitable assumptions on the variable exponents and the diffusion term.

Center manifolds for rough partial differential equations

We prove a center manifold theorem for rough partial differential equations (rough PDEs). The class of rough PDEs we consider contains as a key subclass reaction-diffusion equations driven by nonlinear multiplicative noise, where the stochastic forcing is given by a γ-Hölder rough path, for γ∈(1∕3,1∕2]. Our proof technique relies upon the theory of rough paths and analytic semigroups in combination with a discretized Lyapunov-Perron-type method in a suitable scale of interpolation spaces. The resulting center manifold is a random manifold in the sense of the theory of random dynamical systems (RDS). We also illustrate our main theorem for reaction-diffusion equations as well as for the Swift-Hohenberg equation.

Constructive finite-dimensional boundary control of stochastic 1D parabolic PDEs

Recently, a constructive method for the finite-dimensional observer-based control of deterministic parabolic PDEs was suggested by employing a modal decomposition approach. In this paper, for the first time we extend this method to the stochastic 1D heat equation with nonlinear multiplicative noise. We consider the Neumann actuation and study the observer-based as well as the state-feedback controls via the modal decomposition approach. We employ either trigonometric or polynomial dynamic extension. For observer-based control we consider a noisy boundary measurement. First, we show the well-posedness of strong solutions to the closed-loop systems. Then by suggesting a direct Lyapunov method and employing Itô’s formula, we provide mean-square L2 exponential stability analysis of the full-order closed-loop system, leading to linear matrix inequality (LMI) conditions for finding the observer dimension and as large as possible noise intensity bound for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity and large enough observer dimension (for observer-based control). We further show that in the case of state-feedback and linear noise, the system is always stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system with deterministic measurement. Numerical simulations are carried out to illustrate the efficiency of our method. For both state-feedback and observer-based controls, the trigonometric extension always allows for a larger noise than the polynomial one in the example.

Weak mean random attractors for non-local random and stochastic reaction–diffusion equations

In this paper, we prove the existence of weak pullback mean random attractors for a non-local stochastic reaction–diffusion equation with a nonlinear multiplicative noise. The existence and uniqueness of solutions and weak pullback mean random attractors are also established for a deterministic non-local reaction–diffusion equations with random initial data.

Mean-square invariant manifolds for ill-posed stochastic evolution equations driven by nonlinear noise

This paper discerns the invariant manifold of a class of ill-posed stochastic evolution equations driven by a nonlinear multiplicative noise. To be more precise, we establish the existence of mean-square random unstable invariant manifold and only mean-square stable invariant set. Due to the lack of the Hille-Yosida condition, we construct a modified variation of constants formula by the resolvent operator. With the price of imposing an unusual condition involving a non-decreasing map, we set up the Lyapunov-Perron method and derive the required estimates. We also emphasize that the Lyapunov-Perron map in the forward time loses the invariant due to the adaptedness, we alternatively establish the existence of mean-square random stable sets.

Constructive method for boundary control of stochastic 1D parabolic PDEs

Recently, qualitative methods for finite-dimensional boundary state-feedback control were introduced for stochastic 1D parabolic PDEs. In this paper, we present constructive and efficient design conditions for state-feedback control of stochastic 1D heat equations driven by a nonlinear multiplicative noise. We consider the Neumann actuation and apply modal decomposition with either trigonometric or polynomial dynamic extension. The controller design employs a finite number of comparatively unstable modes. We provide mean-square L2 stability analysis of the full-order closed-loop system, where we employ Itô’s formula, leading to linear matrix inequality (LMI) conditions for finding the controller gain and as large as possible noise intensity for the mean-square stabilizability. We prove that the LMIs are always feasible for small enough noise intensity. We further show that in the case of linear multiplicative noise, the system is stabilizable for noise intensities that guarantee the stabilizability of the stochastic finite-dimensional part of the closed-loop system. Numerical simulations illustrate the efficiency of our method.

Wong–Zakai approximations and random attractors of non-autonomous stochastic discrete complex Ginzburg–Landau equations

In this paper, we consider the Wong–Zakai approximations and random attractors for stochastic discrete complex Ginzburg–Landau equations driven by nonlinear multiplicative noise. We establish the existence of the random attractor for the approximate system. For linear multiplicative noise or additive noise, we study the convergence of the solutions and the upper semicontinuity of random attractors for the approximate equation when the perturbation parameters tend to zero.

Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE

In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces Hs(T)(s>32) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyöngy–Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation.

The existence and exponential behavior of solutions to time fractional stochastic delay evolution inclusions with nonlinear multiplicative noise and fractional noise

This article is devoted to study time fractional stochastic evolution inclusions with infinite delays driven by a nonlinear multiplicative noise and a fractional Brownian motion with Hurst parameter \begin{document}$ H\in(\frac{1}{2},1) $\end{document} . First of all, we investigate the local and global existence of mild solutions to such evolution inclusions by using the fractional resolvent operator theory and some new results on the measure of noncompactness for the stochastic integral term. Further, we prove that every mild solution decays exponentially to zero in the sense of mean-square topology.

Ergodicity for Stochastic Porous Media Equations with Multiplicative Noise

The long time behavior of solutions to stochastic porous media equations with nonlinear multiplicative noise on bounded domains with Dirichlet boundary data is studied. Based on weighted $L^{1}$-es...

A Stochastic Optimal Regulator for a Class of Nonlinear Systems

This work investigates an optimal control problem for a class of stochastic differential bilinear systems, affected by a persistent disturbance provided by a nonlinear stochastic exogenous system (nonlinear drift and multiplicative state noise). The optimal control problem aims at minimizing the average value of a standard quadratic-cost functional on a finite horizon. It has been supposed that neither the state of the system nor the state of the exosystem is directly measurable (incomplete information case). The approach is based on the Carleman embedding, which allows to approximate the nonlinear stochastic exosystem in the form of a bilinear system (linear drift and multiplicative noise) with respect to an extended state that includes the state Kronecker powers up to a chosen degree. This way the stochastic optimal control problem may be restated in a bilinear setting and the optimal solution is provided among all the affine transformations of the measurements. The present work is a nontrivial extension of previous work of the authors, where the Carleman approach was exploited in a framework where only additive noises had been conceived for the state and for the exosystem. Numerical simulations support theoretical results by showing the improvements in the regulator performances by increasing the order of the approximation.

Stochastic compressible Euler equations and inviscid limits

We prove the existence of a unique local strong solution to the stochastic compressible Euler system with nonlinear multiplicative noise. This solution exists up to a positive stopping time and is strong in both the PDE and probabilistic sense. Based on this existence result, we study the inviscid limit of the stochastic compressible Navier–Stokes system. As the viscosity tends to zero, any sequence of finite energy weak martingale solutions converges to the compressible Euler system.

The resonant behavior in the oscillator with double fractional-order damping under the action of nonlinear multiplicative noise

We study stochastic resonance (SR) in an oscillator with nonlinear noise, fractional-order external damping, and fractional-order intrinsic damping. Using a moment equation, we derive the exact analytical expression of the output amplitude and find that fluctuations in the output amplitude are non-monotonic. Using numerical simulations we verify the accuracy of this analytical result. We find (i) that nonlinear noise plays a key role in system behavior and that the resonance of the output amplitude is diverse when there is nonlinear noise, (ii) that the order of the fractional-order damping strongly impacts resonant intensity and that the impact on resonant intensity of fractional-order external damping is opposite that of fractional-order intrinsic damping, and (iii) that the evolution of the output amplitude versus the frequency of the external periodic force exhibits three behaviors: a resonance with one peak, a resonance with one peak and one valley, and a resonance with one valley.

Existence of martingale solutions and the incompressible limit for stochastic compressible flows on the whole space

We give an existence and asymptotic result for the so-called finite energy weak martingale solution of the compressible isentropic Navier–Stokes system driven by some random force in the whole spatial region. In particular, given a general nonlinear multiplicative noise, we establish the convergence to the incompressible system as the Mach number, representing the ratio between the average flow velocity and the speed of sound, approaches zero.