Degenerate Noise Research Articles

Mixing via controllability for randomly forced nonlinear dissipative PDEs

We continue our study of the problem of mixing for a class of PDEs with very degenerate noise. As we established earlier, the uniqueness of stationary measure and its exponential stability in the dual-Lipschitz metric holds under the hypothesis that the unperturbed equation has exactly one globally stable equilibrium point. In this paper, we relax that condition, assuming only global controllability to a given point. It is proved that the uniqueness of a stationary measure and convergence to it are still valid, whereas the rate of convergence is not necessarily exponential. The result is applicable to randomly forced parabolic-type PDEs, provided that the deterministic part of the external force is in general position, ensuring a regular structure for the attractor of the unperturbed problem. The proof uses a new idea that reduces the verification of a stability property to the investigation of a conditional random walk.

Existence and uniqueness of invariant measures of 3D stochastic MHD-α model driven by degenerate noise

ABSTRACT In this paper, we establish the existence and uniqueness of invariant measures of the 3D stochastic magnetohydrodynamic-α model (MHD-α) driven by degenerate additive noise. We firstly study the Feller property of solutions and establish the existence of invariant measures by utilizing the classical Krylov–Bogoliubov theorem. Then, we prove the uniqueness of invariant measures for the corresponding transition semigroup by utilizing the notion of asymptotic strong Feller proposed by Hairer and Mattingly [Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann Math (2). 2006;164(3):993–1032]. The proof not only requires the investigation of degenerate noise, but also the study of highly nonlinear, unbounded drifts.

Ergodicity and exponential mixing of the real Ginzburg-Landau equation with a degenerate noise

In this paper, we establish the existence, uniqueness and attraction properties of an invariant measure for the real Ginzburg-Landau equation in the presence of a degenerate stochastic forcing acting only in four directions. The main challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. To achieve this, we propose a condition which only requires four noises.

Exponential mixing for a class of dissipative PDEs with bounded degenerate noise

We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dual-Lipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a random force which affects only a few Fourier modes and belongs to a certain class of random processes. We assume that the nonlinear PDE in question is well posed, its nonlinearity is non-degenerate in the sense of the control theory, and the random force is a regular and bounded function of time which satisfies some decomposability and observability hypotheses. This class of forces includes random Haar series, where coefficients for high Haar modes decay sufficiently fast. In particular, the result applies to the 2D Navier–Stokes system and the nonlinear complex Ginzburg–Landau equations. The proof of the abstract theorem uses the coupling method, enhanced by the Newton–Kantorovich–Kolmogorov fast convergence.

Ergodic BSDEs with Multiplicative and Degenerate Noise

In this paper we study an Ergodic Markovian BSDE involving a forward process $X$ that solves an infinite dimensional forward stochastic evolution equation with multiplicative and possibly degenerate diffusion coefficient. A concavity assumption on the driver allows us to avoid the typical quantitative conditions relating the dissipativity of the forward equation and the Lipschitz constant of the driver. Although the degeneracy of the noise has to be of a suitable type we can give a stochastic representation of a large class of Ergodic HJB equations; morever our general results can be applied to get the synthesis of the optimal feedback law in relevant examples of ergodic control problems for SPDEs.

Ergodicity of two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise

The current paper is devoted to ergodicity of the stochastic two-dimensional primitive equations of large scale ocean in geophysics driven by degenerate noise. The central challenge is to prove the asymptotically strong Feller property of the probability transition semigroups.

Exponential mixing of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

The current paper is devoted to stochastic Ginzburg–Landau–Newell equation with degenerate stochastic forcing. First, we establish a type of gradient inequality, which is also essential to proving asymptotic strong Feller property. Then, we prove that the corresponding dynamical system possesses a strong type of Lyapunov structure and is of a relatively weak form of irreducibility. Finally, we prove that the corresponding Markov semigroup possesses a unique, exponentially mixing invariant measure.

Invariant behavior of stochastic atmosphere-ocean model with degenerate noise

This work is concerned with the invariant behavior of a stochastic atmosphere-ocean model with the degenerate noise, which describes the climate and geophysical phenomena. Through introducing a transform to deal with the complicated boundary conditions, first, the existence of the invariant measure is established. Since the degenerate noise only drives the latitudinal atmosphere surface temperature, it then proposes to combine the strong Feller property with the asymptotically strong Feller and the irreducibility to verify the uniqueness of the invariant measure. It further implies the ergodicity of the system.

Ergodicity and spike rate for stochastic FitzHugh–Nagumo neural model with periodic forcing

Abstract We discuss ergodicity on a Poincare section and estimate the average spike rate for a time periodically forced stochastic FitzHugh–Nagumo model with degenerate noise. Stochastic FitzHugh–Nagumo (SFHN) model is a prototype stochastic neural oscillator, describing the generation and propagation of action potentials or spikes in an excitable neuron at the intracellular level. Neuronal spikes play significant role in neural information coding of various nervous systems, they are described in terms of an infinitesimal probability that spikes occur, known as spike rate. Estimation of this spike rate is a subtle task for time continuous stochastic processes such as solutions of SFHN model and, in particular, time periodically forced SFHN model. Using the regularity of the ergodic periodic measure, we estimate the average spike rate in terms of the probability density of two-point motions of the membrane potential via Rice’s formula.

Exponential mixing for stochastic 3D fractional Leray-[formula omitted] model with degenerate multiplicative noise

In this work we establish the exponential mixing property for stochastic 3D fractional Leray-α model with degenerate noise. The main result is also applicable to stochastic 3D hyperviscous Navier–Stokes equations and stochastic 3D critical Leray-α model.

Ergodicity of 3D Leray-α model with fractional dissipation and degenerate stochastic forcing

By using the asymptotic coupling method, the asymptotic log-Harnack inequality is established for the transition semigroup associated to the 3D Leray-[Formula: see text] model with fractional dissipation driven by highly degenerate noise. As applications, we derive the asymptotic strong Feller property and ergodicity for the stochastic 3D Leray-[Formula: see text] model with fractional dissipation, which is the stochastic 3D Navier–Stokes equation regularized through a smoothing kernel of order [Formula: see text] in the nonlinear term and a [Formula: see text]-fractional Laplacian. The main results can be applied to the classical stochastic 3D Leray-[Formula: see text] model ([Formula: see text]), stochastic 3D hyperviscous Navier–Stokes equation ([Formula: see text]) and stochastic 3D critical Leray-[Formula: see text] model ([Formula: see text]).

Invariant measure of stochastic fractional Burgers equation with degenerate noise on a bounded interval

This work is concerned with the invariant measure of a stochastic fractional Burgers equation with degenerate noise on one dimensional bounded domain. Due to the disturbance and influence of the fractional Laplacian operator on a bounded interval interacting with the degenerate noise, the study of the system becomes more complicated. In order to get over the difficulties caused by the fractional Laplacian operator, the usual Hilbert space does not fit the system, we introduce an appropriate weighted space to study it. Meanwhile, we apply the asymptotically strong Feller property instead of the usually strong Feller property to overcome the trouble caused by the degenerate noise, the corresponding Malliavin operator is not invertible. We finally derive the uniqueness of the invariant measure which further implies the ergodicity of the stochastic system.

Ergodic control of infinite-dimensional stochastic differential equations with degenerate noise

This paper is devoted to the study of the asymptotic behaviour of the value functions of both finite and infinite horizon stochastic control problems and to the investigation of their relationship with suitable stochastic ergodic control problems. Our methodology is based only on probabilistic techniques, as, for instance, the so-called randomisation of the control method, thus avoiding completely analytical tools from the theory of viscosity solutions. We are then able to treat with the case where the state process takes values in a general (possibly infinite-dimensional) real separable Hilbert space, and the diffusion coefficient is allowed to be degenerate.

Thermoviscous fluid flow modes in a plane nonisothermal layer

This paper deals with 3D flow of thermoviscous fluid in the low compressibility approximation within a cubic-shaped domain enclosed between two flat plates with different temperatures. For two other directions, the problem statement assigns periodic boundary conditions, while the steady pressure drop is sustained for the head flow direction. Such formulation allows to trace the evolution of initial disturbances imposed on the main flow depending on perturba-tion properties. In this case, we consider a degenerate one-dimensional divergence-free noise that is modified by a special correlation filter. When the divergent noise is generated, the solenoid nature of random velocity field must be restored. The simulation demonstrates that random disturbance field development leads to two different scenarios: for the first low-amplitude case, the velocity profile loses initial inflection point and its flowrate increases by 1.5–1.6 times, but for the second one, the flow turbulization occurs destroying the flow core and decreasing the flowrate. In both outcomes, the transition to a steady flow mode in terms of either stationary velocity fields or statistical averag-es takes place for a long interval: up to t~200 dimensionless time units. The analysis of simulated flow is based on integral kinetic energy curves and enstrophy and also via spatial averaging of the obtained data arrays.

Ergodicity of a Galerkin approximation of three-dimensional magnetohydrodynamics system forced by a degenerate noise

Magnetohydrodynamics system consists of a coupling of the Navier-Stokes and Maxwell's equations and is most useful in studying the motion of electrically conducting fluids. We prove the existence of a unique invariant, and consequently ergodic, measure for the Galerkin approximation system of the three-dimensional magnetohydrodynamics system. The proof is inspired by those of [E. Weinan and J.C. Mattingly, Ergodicity for the Navier-Stokes equation with degenerate random forcing: Finitedimensional approximation, Comm. Pure Appl. Math. LIV (2001), pp. 1386–1402; M. Romito, Ergodicity of the finite dimensional approximation of the 3D Navier-Stokes equations forced by a degenerate noise, J. Stat. Phys. 114 (2004), pp. 155–177] on the Navier-Stokes equations; however, computations involve significantly more complications due to the coupling of the velocity field equations with those of magnetic field that consists of four non-linear terms.

Persistence in Stochastic Lotka-Volterra Food Chains with Intraspecific Competition.

This paper is devoted to the analysis of a simple Lotka-Volterra food chain evolving in a stochastic environment. It can be seen as the companion paper of Hening and Nguyen (J Math Biol 76:697-754, 2018b) where we have characterized the persistence and extinction of such a food chain under the assumption that there is no intraspecific competition among predators. In the current paper, we focus on the case when all the species experience intracompetition. The food chain we analyze consists of one prey and [Formula: see text] predators. The jth predator eats the [Formula: see text]st species and is eaten by the [Formula: see text]st predator; this way each species only interacts with at most two other species-the ones that are immediately above or below it in the trophic chain. We show that one can classify, based on the invasion rates of the predators (which we can determine from the interaction coefficients of the system via an algorithm), which species go extinct and which converge to their unique invariant probability measure. We obtain stronger results than in the case with no intraspecific competition because in this setting we can make use of the general results of Hening and Nguyen (Ann Appl Probab 28:1893-1942, 2018a). Unlike most of the results available in the literature, we provide an in-depth analysis for both non-degenerate and degenerate noise. We exhibit our general results by analyzing trophic cascades in a plant-herbivore-predator system and providing persistence/extinction criteria for food chains of length [Formula: see text].

Noise dependent synchronization of a degenerate SDE

We provide an example of an SDE with degenerate additive noise where synchronization depends on the strength of noise and the number of directions in which the noise acts. Here, synchronization means that the weak random attractor consists of a single random point. Indicated by a change of sign of the top Lyapunov exponent, we prove synchronization and no (weak) synchronization respectively.

Ergodicity of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

Ergodicity of 2D stochastic Ginzburg–Landau–Newell equations driven by degenerate noise

Nonlinear filtering with degenerate noise

This paper studies a new type of filtering problem, where the diffusion coefficient of the observation noise is strictly positive only in the interior of the bounded interval where observation takes its values. We derive a Zakai and a Kushner–Stratonovich equation, and prove uniqueness of the measure–valued solution of the Zakai equation.

Amplitude equation with quintic nonlinearities for the generalized Swift-Hohenberg equation with additive degenerate noise

In this paper, we are interested in the approximation of a stochastic generalized Swift-Hohenberg equation with quadratic and cubic nonlinearity by using the natural separation of time-scales near a change of stability. The main results show that the behavior of the SPDE is well approximated by a stochastic ordinary differential equation describing the amplitude of the dominant mode. The cubic and the quadratic nonlinearities lead to cubic nonlinearities of opposite sign. Here we study the interesting case, where both contributions cancel and in the right scaling a quintic nonlinearity emerges in the amplitude equation. Also, we give a brief indication of how the effect of additive degenerate noise (i.e. noise that does not act directly to the dominant mode) might lead to the stabilization of the trivial solution.