Small Multiplicative Noise Research Articles

Large deviation principle for pseudo-monotone evolutionary equation

. In this article, we explore Freidlin-Wentzell’s large deviation principle for a stochastic partial differential equation with the nonlinear diffusion-convection operator in divergence form satisfying p-type growth, involving coercivity assumptions, perturbed by small multiplicative Brownian noise. We use the weak convergence method to prove the Laplace principle, which is equivalent to the large deviation principle in our framework.

Stochastic evolutionary p-Laplace equation: Large deviation principles and transportation cost inequality

We establish a large deviation principle for the strong solution of the evolutionary p-Laplace equation driven by small multiplicative Brownian noise, where the weak convergence approach plays a key role. Moreover, by using the Girsanov transformation along with the L1-contraction approach, we show the quadratic transportation cost inequality for the strong solution to the underlying problem.

Long Time Behavior of Stochastic NLS with a Small Multiplicative Noise

Long Time Behavior of Stochastic NLS with a Small Multiplicative Noise

Large deviations for the two-time-scale stochastic convective Brinkman-Forchheimer equations

The convective Brinkman-Forchheimer (CBF) equations are employed to characterize the motion of incompressible fluid in a saturated porous medium. This work investigates the small noise asymptotic of two-time-scale stochastic CBF equations in two and three dimensional bounded domains. More precisely, we establish a Wentzell-Freidlin type large deviation principle for stochastic partial differential equations that have slow and fast time-scales. The slow component is the stochastic CBF equations in two or three dimensions perturbed by a small multiplicative Gaussian noise, while the fast component is a stochastic reaction-diffusion equation with damping. The results are obtained by using a variational method (based on weak convergence approach) developed by Budhiraja and Dupuis, Khasminkii's time discretization approach and stopping time arguments. In particular, the findings from this study are also applicable to two-dimensional stochastic Navier-Stokes equations.

Metastability and exit problems for systems of stochastic reaction–diffusion equations

In this paper we develop a metastability theory for a class of stochastic reaction–diffusion equations exposed to small multiplicative noise. We consider the case where the unperturbed reaction–diffusion equation features multiple asymptotically stable equilibria. When the system is exposed to small stochastic perturbations, it is likely to stay near one equilibrium for a long period of time but will eventually transition to the neighborhood of another equilibrium. We are interested in studying the exit time from the full domain of attraction (in a function space) surrounding an equilibrium and, therefore, do not assume that the domain of attraction features uniform attraction to the equilibrium. This means that the boundary of the domain of attraction is allowed to contain saddles and limit cycles. Our method of proof is purely infinite dimensional, that is, we do not go through finite dimensional approximations. In addition, we address the multiplicative noise case, and we do not impose gradient type of assumptions on the nonlinearity. We prove large deviations logarithmic asymptotics for the exit time and for the exit shape, also characterizing the most probable set of shapes of solutions at the time of exit from the domain of attraction.

Large Deviation Principle for McKean–Vlasov Quasilinear Stochastic Evolution Equations

This paper is devoted to investigating the Freidlin–Wentzell’s large deviation principle for a class of McKean–Vlasov quasilinear SPDEs perturbed by small multiplicative noise. We adopt the variational framework and the modified weak convergence criteria to prove the Laplace principle for McKean–Vlasov type SPDEs, which is equivalent to the large deviation principle. Moreover, we do not assume any compactness condition of embedding in the Gelfand triple to handle both the cases of bounded and unbounded domains in applications. The main results can be applied to various McKean–Vlasov type SPDEs such as distribution dependent stochastic porous media type equations and stochastic p-Laplace type equations.

Dominant dynamics for a class of singularly perturbed stochastic partial differential equations with quadratic nonlinearities and random Neumann boundary conditions.

This work concerns the effective approximation for a class of singularly perturbed stochastic partial differential equations driven by a sufficiently small multiplicative noise with quadratic nonlinearities and random Neumann boundary conditions. By splitting the solution into two parts in the finite dimension kernel space and its complement space with some suitable multi-scale argument, it derives rigorously the dominant dynamics, which captures the essential dynamics of the original system as a singular parameter is enough small.

Large deviation principle for stochastic Burgers type equation with reflection

<p style='text-indent:20px;'>In this paper, we establish a large deviation principle for stochastic Burgers type equation with reflection perturbed by the small multiplicative noise. The main difficulties come from the highly non-linear coefficient and the singularity caused by the reflection. Here, we adopt a new sufficient condition for the weak convergence criteria, which is proposed by Matoussi, Sabbagh and Zhang [<xref ref-type="bibr" rid="b14">14</xref>].</p>

Decay of the stochastic linear Schr\xf6dinger equation in d ge 3 with small multiplicative noise

We give decay estimates of the solution to the linear Schrödinger equation in dimension d ge 3 with a small noise which is white in time and colored in space. As a consequence, we also obtain certain asymptotic behaviour of the solution. The proof relies on the bootstrapping argument used by Journé–Soffer–Sogge for decay of deterministic Schrödinger operators.

Large deviation principles for first-order scalar conservation laws with stochastic forcing

In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.

Large deviations for stochastic porous media equations

In this paper, we establish the Freidlin-Wentzell type large deviation principle for porous medium-type equations perturbed by small multiplicative noise. The porous medium operator $\Delta (|u|^{m-1}u)$ is allowed. Our proof is based on weak convergence approach.

Stability of Traveling Waves for Systems of Reaction-Diffusion Equations with Multiplicative Noise

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system ret...

Travelling waves for reaction–diffusion equations forced by translation invariant noise

Inspired by applications, we consider reaction–diffusion equations on R that are stochastically forced by a small multiplicative noise term that is white in time, coloured in space and invariant under translations. We show how these equations can be understood as a stochastic partial differential equation (SPDE) forced by a cylindrical Q-Wiener process and subsequently explain how to study stochastic travelling waves in this setting. In particular, we generalize the phase tracking framework that was developed in Hamster and Hupkes (2018,2019) for noise processes driven by a single Brownian motion. The main focus lies on explaining how this framework naturally leads to long term approximations for the stochastic wave profile and speed. We illustrate our approach by two fully worked-out examples, which highlight the predictive power of our expansions.

Large deviations for the 2-D derivative Ginzburg–Landau equation with multiplicative noise

This paper proves a Wentzell–Freidlin type large deviation principle for the 2-D derivative Ginzburg–Landau equation perturbed by a small multiplicative noise, based on the Laplace principle and weak convergence approach.

Large deviations for stochastic 3D Leray-<inline-formula><tex-math id="M1">\begin{document}$ \alpha $\end{document}</tex-math></inline-formula> model with fractional dissipation

In this paper we establish the Freidlin-Wentzell's large deviation principle for stochastic 3D Leray-$ \alpha $ model with general fractional dissipation and small multiplicative noise. This model is the stochastic 3D Navier-Stokes equations regularized through a smoothing kernel of order $ \theta_1 $ in the nonlinear term and a $ \theta_2 $-fractional Laplacian. The main result generalizes the corresponding LDP result of the classical stochastic 3D Leray-$ \alpha $ model ($ \theta_1 = 1 $, $ \theta_2 = 1 $), and it is also applicable to the stochastic 3D hyperviscous Navier-Stokes equations ($ \theta_1 = 0 $, $ \theta_2\geq\frac{5}{4} $) and stochastic 3D critical Leray-$ \alpha $ model ($ \theta_1 = \frac{1}{4} $, $ \theta_2 = 1 $).

Stability of Traveling Waves for Reaction-Diffusion Equations with Multiplicative Noise

We consider reaction-diffusion equations that are stochastically forced by a small multiplicative noise term. We show that spectrally stable traveling wave solutions to the deterministic system retain their orbital stability if the amplitude of the noise is sufficiently small. By applying a stochastic phase shift together with a time transform, we obtain a semilinear Stochastic partial differential equation that describes the fluctuations from the primary wave. We subsequently develop a semigroup approach to handle the nonlinear stability question in a fashion that is closely related to modern deterministic methods.

Mean extinction time and stability for a metapopulation system subjected to correlated Gaussian and non-Gaussian noises

In this paper, we investigate the shift between the stable state of a large density and the extinction state, the mean extinction time for a metapopulation system induced by the associated Gaussian additive noise and non-Gaussian multiplicative noise. By applying the approximate Fokker–Planck equation, the fast descent method and the path integral approach, we obtain the expressions of the effective potential function and the mean extinction time. The numerical results show that the additive noise can damage the stability of the system, while the associated noise and the noise correlation times τ0 and τ can strengthen the stability of the population system. In the meantime, the maximum of the mean first-passage time appears in the meta-population system due to the impacts of different types of noises and their correlation time terms. It can effectively extend the extinction time of the metapopulation to enhance the strength of the correlation noise and two terms of noise correlation time. Conversely, the additive noise plays a negative role in preventing the extinction of the population. On the other side, a proper small multiplicative noise intensity can produce the positive effect on lengthening the life of the metapopulation.

Random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise

ABSTRACTThe aim of this article is to study the asymptotical behavior, in terms of upper semi-continuous property of attractor, for small multiplicative noise of the three-dimensional planetary geostrophic equations of large-scale ocean circulation. In this article, we establish the existence of a random attractor for the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise by verifying the pullback flattening property and prove that the random attractor of the three-dimensional planetary geostrophic equations of large-scale ocean circulation with small multiplicative noise converges to the global attractor of the unperturbed three-dimensional planetary geostrophic equations of large-scale ocean circulation when the parameter of the perturbation tends to zero.

Persistence of Homoclinic Orbits under White Noise Perturbation

In this paper, we study the persistence of homoclinic orbits for some second order ordinary differential equations subject to small multiplicative white noise perturbation. We obtain a path-independent criterion, which can be thought of as a generalization of the Melnikov integral, to ensure the persistence of homoclinic orbits for almost all sample paths. The proof is based on invariant manifold theory and Melnikov-type analysis.

Large deviations for stochastic 3D cubic Ginzburg–Landau equation with multiplicative noise

Abstract This paper considers the large deviation principle for the stochastic 3D cubic Ginzburg–Landau equation perturbed by a small multiplicative noise. Using the weak convergence approach, we establish a large deviation principle of Freidlin–Wentzell type by proving a Laplace principle.