Degenerate Noise Research Articles

Degenerate Nucleation in the Cahn--Hilliard--Cook Model

Phase separation in metal alloys is an important pattern forming physical process with applications in materials science, both for understanding materials structure and for the design of new materials. The Cahn--Hilliard equation is a deterministic model for the dynamics of alloys which has proven to be fundamental for the understanding of several types of phase separation behavior. However, stochastic effects occur in any physical experiment and thus need to be incorporated into models. While white noise is one standardly chosen option, it is not immediately clear in which way the noise characteristics affect the resulting patterns. In this paper we study the effects of not necessarily small colored noise on pattern formation in a stochastic Cahn--Hilliard model in the nucleation regime. More precisely, we focus on degenerate noise which acts either on isolated eigenmodes or with a well-defined spatial wavelength. Our studies show that the types of resulting patterns depend critically on the spatial nois...

Large-time and small-ball asymptotics for quadratic functionals of Gaussian diffusions

Using asymptotic analysis of the Laplace transform, we establish almost sure divergence of certain integrals and derive logarithmic asymptotic of small ball probabilities for quadratic forms of Gaussian diffusion processes. The large time behavior of the quadratic forms exhibits little dependence on the drift and diffusion matrices or the initial conditions, and, if the noise driving the equation is not degenerate, then similar universality also holds for small ball probabilities. On the other hand, degenerate noise leads to a variety of different asymptotics of small ball probabilities, including unexpected influence of the initial conditions.

Asymptotic properties of stochastic Cahn–Hilliard equation with singular nonlinearity and degenerate noise

We consider a stochastic partial differential equation with a logarithmic nonlinearity with singularities at 1 and −1 and a constraint of conservation of the space average. The equation, driven by a trace-class space–time noise, contains a bi-Laplacian in the drift. We obtain existence of solution for equation with polynomial approximation of the nonlinearity. Tightness of this sequence of approximations is proved, leading to a limit transition semi-group. We study the asymptotic properties of this semi-group, showing the existence and uniqueness of invariant measure, asymptotic strong Feller property and topological irreducibility.

Sub and supercritical stochastic quasi-geostrophic equation

In this paper, we study the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^{2}$ for general parameter $\alpha\in(0,1)$ and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all $\alpha\in(0,1)$. In the subcritical case $\alpha>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions. Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven by possibly degenerate noise. The law of large numbers for the solution of the stochastic quasi-geostrophic equations in the subcritical case is also established. In the case of nondegenerate noise and $\alpha>2/3$ in addition exponential ergodicity is proved.

Loss of regularity for Kolmogorov equations

The celebrated H\"{o}rmander condition is a sufficient (and nearly necessary) condition for a second-order linear Kolmogorov partial differential equation (PDE) with smooth coefficients to be hypoelliptic. As a consequence, the solutions of Kolmogorov PDEs are smooth at all positive times if the coefficients of the PDE are smooth and satisfy H\"{o}rmander's condition even if the initial function is only continuous but not differentiable. First-order linear Kolmogorov PDEs with smooth coefficients do not have this smoothing effect but at least preserve regularity in the sense that solutions are smooth if their initial functions are smooth. In this article, we consider the intermediate regime of nonhypoelliptic second-order Kolmogorov PDEs with smooth coefficients. The main observation of this article is that there exist counterexamples to regularity preservation in that case. More precisely, we give an example of a second-order linear Kolmogorov PDE with globally bounded and smooth coefficients and a smooth initial function with compact support such that the unique globally bounded viscosity solution of the PDE is not even locally H\"{o}lder continuous. From the perspective of probability theory, the existence of this example PDE has the consequence that there exists a stochastic differential equation (SDE) with globally bounded and smooth coefficients and a smooth function with compact support which is mapped by the corresponding transition semigroup to a function which is not locally H\"{o}lder continuous. In other words, degenerate noise can have a roughening effect. A further implication of this loss of regularity phenomenon is that numerical approximations may converge without any arbitrarily small polynomial rate of convergence to the true solution of the SDE. More precisely, we prove for an example SDE with globally bounded and smooth coefficients that the standard Euler approximations converge to the exact solution of the SDE in the strong and numerically weak sense, but at a rate that is slower then any power law.

Amplitude equation for the stochastic reaction‐diffusion equations with random Neumann boundary conditions

In this paper, we consider a quite general class of reaction‐diffusion equations with cubic nonlinearities and with random Neumann boundary conditions. We derive rigorously amplitude equations, using the natural separation of time‐scales near a change of stability and investigate whether additive degenerate noise and random boundary conditions can lead to stabilization of the solution of the stochastic partial differential equation or not. The nonlinear heat equation (Ginzburg–Landau equation) is used to illustrate our result. Copyright © 2015 John Wiley & Sons, Ltd.

Exponential mixing of 2D SDEs forced by degenerate Lévy noises

We modify the coupling method established in Shirikyan (Exponential mixing for randomly forced partial differential equations: method of coupling, Springer, New York, 2008) and Shirikyan (J Math Fluid Mech 6(2):169–193, 2004) and develop a technique to prove the exponential mixing of a 2D stochastic system forced by degenerate Levy noises. In particular, these Levy noises include α-stable noises (0 < α < 2). Thanks to the stimulating discussion (Nersesyan in Private communication 2011), this technique is promising to study the exponential mixing problem of SPDEs driven by degenerate symmetric α-stable noises.

Calibrating the exponential Ornstein–Uhlenbeck multiscale stochastic volatility model

This paper demonstrates a tractable and efficient way of calibrating a multiscale exponential Ornstein–Uhlenbeck stochastic volatility model including a correlation between the asset return and its volatility. As opposed to many contributions where this correlation is assumed to be null, this framework allows one to describe the leverage effect widely observed in equity markets. The resulting model is non-exponential and driven by a degenerate noise, thus requiring a high level of care in designing the estimation algorithm. The way this difficulty is overcome provides guidelines concerning the development of an estimation algorithm in a non-standard framework. The authors propose using a block-type expectation maximization algorithm along with particle smoothing. This method results in an accurate calibration process able to identify up to three timescale factors. Furthermore, we introduce an intuitive heuristic which can be used to choose the number of factors.

Numerical study of amplitude equations for SPDEs with degenerate forcing

In this paper, we give a review on rigorous and numerical results for amplitude equations. We focus on the Swift–Hohenberg equation and the Burgers’ equation in order to determine the quality of the approximation and the impact of degenerate noise on the approximating equation.

The stationary distribution of the facultative population model with a degenerate noise

In this paper, we consider the stationary distribution of the facultative population model with a degenerate noise. The contributions of this paper lie in: (a) providing sufficient conditions which allow the noise intensity matrix to be degenerate, and, in particular, guarantee the existence and uniqueness of the stationary distribution of our model; (b) discussing the property of positive recurrence of the model and revealing that the associated transition probability function converges exponentially to the unique stationary distribution; (c) showing the integral equation that the Laplace transform of the stationary distribution satisfies.

Additive noise effects in active nonlinear spatially extended systems

We examine the effects of pure additive noise on spatially extended systems with quadratic nonlinearities. We develop a general multi-scale theory for such systems and apply it to the Kuramoto–Sivashinsky equation as a case study. We first focus on a regime close to the instability onset (primary bifurcation), where the system can be described by a single dominant mode. We show analytically that the resulting noise in the equation describing the amplitude of the dominant mode largely depends on the nature of the stochastic forcing. For a highly degenerate noise, in the sense that it is acting on the first stable mode only, the amplitude equation is dominated by a pure multiplicative noise, which in turn induces the dominant mode to undergo several critical state transitions and complex phenomena, including intermittency and stabilisation, as the noise strength is increased. The intermittent behaviour is characterised by a power-law probability density and the corresponding critical exponent is calculated rigorously by making use of the first-passage properties of the amplitude equation. On the other hand, when the noise is acting on the whole subspace of stable modes, the multiplicative noise is corrected by an additive-like term, with the eventual loss of any stabilised state. We also show that the stochastic forcing has no effect on the dominant mode dynamics when it is acting on the second stable mode. Finally, in a regime which is relatively far from the instability onset so that there are two unstable modes, we observe numerically that when the noise is acting on the first stable mode, both dominant modes show noise-induced complex phenomena similar to the single-mode case.

Exponential Mixing of the 3D Stochastic Navier-Stokes Equations Driven by Mildly Degenerate Noises

We prove the strong Feller property and exponential mixing for 3D stochastic Navier-Stokes equation driven by mildly degenerate noises (i.e. all but finitely many Fourier modes being forced) via a Kolmogorov equation approach.

Amplitude equations for SPDEs with cubic nonlinearities

For a quite general class of stochastic partial differential equations with cubic nonlinearities, we derive rigorously amplitude equations describing the essential dynamics using the natural separation of timescales near a change of stability. Typical examples are the Swift–Hohenberg equation, the Ginzburg–Landau (or Allen–Cahn) equation and some model from surface growth. We discuss the impact of degenerate noise on the dominant behaviour, and see that additive noise has the potential to stabilize the dynamics of the dominant modes. Furthermore, we discuss higher order corrections to the amplitude equation.

Pathwise Stability of Degenerate Stochastic Evolutions

For linear stochastic evolution equations with linear multiplicative noise, a new method is presented for estimating the pathwise Lyapunov exponent. The method consists of finding a suitable (quadratic) Lyapunov function by means of solving an operator inequality. One of the appealing features of this approach is the possibility to show stabilizing effects of degenerate noise. The results are illustrated by applying them to the examples of a stochastic partial differential equation and a stochastic differential equation with delay. In the case of a stochastic delay differential equation our results improve upon earlier results.

Ergodicity of the 3D stochastic Navier–Stokes equations driven by mildly degenerate noise

We prove that any Markov solution to the 3D stochastic Navier–Stokes equations driven by a mildly degenerate noise (i.e. all but finitely many Fourier modes are forced) is uniquely ergodic. This follows by proving strong Feller regularity and irreducibility.

Infinite Horizon Stochastic Optimal Control Problems with Degenerate Noise and Elliptic Equations in Hilbert Spaces

Semilinear elliptic partial differential equations are solved in a mild sense in an infinite-dimensional Hilbert space. These results are applied to a stochastic optimal control problem with infinite horizon. Applications to controlled stochastic heat and wave equations are given.

Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing

The stochastic 2D Navier-Stokes equations on the torus driven by degenerate noise are studied. We characterize the smallest closed invariant subspace for this model and show that the dynamics restricted to that subspace is ergodic. In particular, our results yield a purely geometric characterization of a class of noises for which the equation is ergodic in L 2 0 (T 2 ). Unlike previous works, this class is independent of the viscosity and the strength of the noise. The two main tools of our analysis are the asymptotic strong Feller property, introduced in this work, and an approximate integration by parts formula. The first, when combined with a weak type of irreducibility, is shown to ensure that the dynamics is ergodic. The second is used to show that the first holds un

Markov solutions for the 3D stochastic Navier–Stokes equations with state dependent noise

We construct a Markov family of solutions for the 3D Navier-Stokes equations perturbed by a non degenerate noise. We improve the result of [3] in two directions. We see that in fact not only a transition semigroup but a Markov family of solutions can be constructed. Moreover, we consider a state dependant noise. Another feature of this work is that we greatly simplify the proofs of [3].

Fault tolerances using toroidal zone plate encryption

We present an analysis of the sensitivity to misalignment, illuminating wavelength change or construction parameters alterations for a zone plate used in an encryption method. The security system is based on a computer generated toroidal phase mask. We use a processor where this phase mask is placed in the Fourier plane of the object to be encrypted. As the original-data recovering is performed by using the conjugate of the encryption mask, a perfect match with the original conditions or the mask characteristics influences not only on the quality of the decrypted image, but also in the success of the method. Our results show that the scheme is an optimized design concerning both data storage and encryption as exhibit less degenerate noise while keeping the security standards of other methods.

Ergodicity of the Finite Dimensional Approximation of the 3D Navier–Stokes Equations Forced by a Degenerate Noise

We prove ergodicity of the finite dimensional approximations of the three dimen- sional Navier-Stokes equations, driven by a random force. The forcing noise acts only on a few modes and some algebraic conditions on the forced modes are found that imply the ergodicity. The convergence rate to the unique invariant measure is shown to be exponential. The uniqueness of statistical steady states for the Navier-Stokes equations is a less known but nevertheless important problem in the mathematical theory of turbulence. The question is completely open in dimension three, mainly because, due to the lack of uniqueness of the equations, there is no way yet to give meaning to the mathematical objects involved in the subject. In the present paper the property of ergodicity is proved for the finite dimensional approxi- mations of the three-dimensional Navier-Stokes equations, driven by a random force. The same problem has been solved in two dimensions by E and Mattingly (2). Such result can have a qualitative interest for the statistical behaviour of an incompressible fluid. Indeed, if the Kolmogorov theory of turbulence is taken into account, one can believe that the cascade of energy, responsible of the transport of the energy through the scales, is effective in the inertial range so that at smaller scales only the dissipation ends up to be relevant. Hence the long-time statistical properties of the fluid can be sufficiently depicted by the low modes of the velocity field. In some sense, if the ultraviolet cut-off is sufficiently large, in order to capture all the important modes, the corresponding invariant measure gives the real behaviour of the fluid. In view of these considerations, the conclusions of the paper can give also both a hint and a possible starting point for the analysis of the infinite dimensional case. We consider a finite dimensional truncation of the three dimensional Navier-Stokes equations, driven by a random force, with periodic boundary conditions. The proof of ergodicity is classical and it is developed in two steps. Firstly we prove that the transition probability densities are regular, by checking that the diffusion operator is hypoelliptic (the Hormander condition). Then we show that the Markov process is irreducible, in the sense that each open set is visited with positive probability at each time. For this aim we study the associated control problem (see Section 6). Irreducibility for the infinite dimensional equations was firstly proved by Flandoli (4), under the assumption that the noise acts on all modes.