Degenerate Noise Research Articles

The complex Ginzburg-Landau equation perturbed by a force localised both in physical and Fourier spaces

In the paper arXiv:1802.03250, a criterion for exponential mixing is established for a class of random dynamical systems. In that paper, the criterion is applied to PDEs perturbed by a noise localised in the Fourier space. In the present paper, we show that, in the case of the complex Ginzburg-Landau (CGL) equation, that criterion can be used to consider even more degenerate noise that is localised both in physical and Fourier spaces. This is achieved by checking that the linearised equation is almost surely approximately controllable. We also study the problem of controllability of the nonlinear CGL equation. Using Agrachev-Sarychev type arguments, we prove an approximate controllability property in the case of a control force which is again localised in physical and Fourier spaces.

Asymptotic Log-Harnack inequality for the 2D Ericksen–Leslie model system with degenerate noise and damping

In this paper, we consider a 2D liquid crystal model with damping and examine some asymptotic behaviors of the strong solution. More precisely, we establish the asymptotic log-Harnack inequality for the transition semigroup associated with a simplified liquid crystal model driven by an additive degenerate noise via the asymptotic coupling method. As consequences of the asymptotic log-Harnack inequality, we derive the gradient estimate, the asymptotic irreducibility, the asymptotic strong Feller property, the asymptotic heat kernel estimate and the ergodicity.

A functional law of the iterated logarithm for weakly hypoelliptic diffusions at time zero

We study the almost sure behavior of solutions of stochastic differential equations (SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hörmander condition but not the strong Hörmander condition. That is, SDEs in which the drift terms must be used in order to conclude hypoellipticity. As a corollary of this result, we obtain the almost sure behavior as time goes to zero of a given direction in the equation, even if noise is not present explicitly in that direction. The techniques used to prove the main results are based on large deviations applied to a non-trivial rescaling of the original system. In concrete examples, we show how to find the proper rescaling to obtain the functional LIL. Furthermore, we apply the main results to the problem of identifying regular points for hypoelliptic diffusions. Consequently, we obtain a control-theoretic criteria for a given point to be regular for the process.

Stochastic Synchronization of Nonlinear Networks With Directed Graphs and Degenerate Noise

Noise is ubiquitous in network systems and has profound impacts on network dynamics. Despite substantial progress in stochastic synchronization of complex networks, most studies limit their focus to nondegenerate noise. However, the influence of degenerate noise on synchronization has not been studied properly. Here, we tackle this challenging problem by introducing a network of nonlinear units with a deterministic communication digraph and a noisy communication digraph. The communication links in the noisy communication digraph are affected by degenerate noise in a relative-dependent manner. Two types of deterministic communication digraphs are considered: 1) the case of strongly connected graphs and 2) the case of graphs containing a spanning tree. By using the algebraic graph theory and the stability theory of stochastic differential equations, algebraic sufficient conditions for both cases are derived, which depend on node dynamics, network structure, and coupling strengths. These criteria demonstrate that strong deterministic communication can compensate for the detrimental effect of degenerate noise, thus ensuring the stochastic synchronization in the almost sure sense. Numerical simulations support and illustrate the proposed theoretical results.

Ergodicity of Galerkin approximations of surface quasi-geostrophic equations and Hall-magnetohydrodynamics system forced by degenerate noise

Ergodicity of Galerkin approximations of surface quasi-geostrophic equations and Hall-magnetohydrodynamics system forced by degenerate noise

Harnack inequality and gradient estimate for functional <i>G</i>-SDEs with degenerate noise

In this paper, the Harnack and shift Harnack inequalities for functional G-SDEs with degenerate noise are derived by the method of coupling by change of measure. Moreover, the gradient estimate for the associated nonlinear semigroup $\bar{P}_t $ is obtained. All of the above results extend the existed results in linear expectation setting.

ISALT: Inference-based schemes adaptive to large time-stepping for locally Lipschitz ergodic systems

<p style='text-indent:20px;'>Efficient simulation of SDEs is essential in many applications, particularly for ergodic systems that demand efficient simulation of both short-time dynamics and large-time statistics. However, locally Lipschitz SDEs often require special treatments such as implicit schemes with small time-steps to accurately simulate the ergodic measures. We introduce a framework to construct inference-based schemes adaptive to large time-steps (ISALT) from data, achieving a reduction in time by several orders of magnitudes. The key is the statistical learning of an approximation to the infinite-dimensional discrete-time flow map. We explore the use of numerical schemes (such as the Euler-Maruyama, the hybrid RK4, and an implicit scheme) to derive informed basis functions, leading to a parameter inference problem. We introduce a scalable algorithm to estimate the parameters by least squares, and we prove the convergence of the estimators as data size increases.</p><p style='text-indent:20px;'>We test the ISALT on three non-globally Lipschitz SDEs: the 1D double-well potential, a 2D multiscale gradient system, and the 3D stochastic Lorenz equation with a degenerate noise. Numerical results show that ISALT can tolerate time-step magnitudes larger than plain numerical schemes. It reaches optimal accuracy in reproducing the invariant measure when the time-step is medium-large.</p>

Asymptotic Log-Harnack Inequality for Stochastic Fractional MHD Equations with Degenerate Noise

<p style='text-indent:20px;'>In this paper we consider the stochastic <inline-formula><tex-math id="M1">\begin{document}$ d $\end{document}</tex-math></inline-formula>-dimensional (<inline-formula><tex-math id="M2">\begin{document}$ d = 2, 3 $\end{document}</tex-math></inline-formula>) magneto-hydrodynamic (MHD) equations with fractional dissipations <inline-formula><tex-math id="M3">\begin{document}$ ((-\Delta)^\alpha $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ (-\Delta)^\beta) $\end{document}</tex-math></inline-formula> driven by degenerate noise. Here <inline-formula><tex-math id="M5">\begin{document}$ \alpha, \beta $\end{document}</tex-math></inline-formula> represent the parameters of the fractional dissipations corresponding to the velocity field and that to the magnetic field respectively. Using the asymptotic coupling method, the asymptotic log-Harnack inequality is established when <inline-formula><tex-math id="M6">\begin{document}$ \alpha\wedge\beta\geq\frac{1}{2}+\frac{d}{4} $\end{document}</tex-math></inline-formula>. As applications, we derive the gradient estimate, asymptotic irreducibility, asymptotic heat kernel estimate and ergodicity for the model.</p>

Irreducibility of Kuramoto-Sivashinsky equation driven by degenerate noise

In this paper, we study irreducibility of Kuramoto-Sivashinsky equation which is driven by an additive noise acting only on a finite number of Fourier modes. In order to obtain the irreducibility, we first investigate the approximate controllability of Kuramoto-Sivashinsky equation driven by a finite-dimensional force, the proof is based on Agrachev-Sarychev type geometric control approach. Next, we study the continuity of solving operator for deterministic Kuramoto-Sivashinsky equation. Finally, combining the approximate controllability with continuity of solving operator, we establish the irreducibility of Kuramoto-Sivashinsky equation.

Asymptotic properties of the 2D stochastic fractional Boussinesq equations driven by degenerate noise

The current paper is devoted to stochastic fractional Boussinesq equations (FBEs) on the Torus T2=[0,2π]×[0,2π] with degenerate random forcing. We first show the existence and pathwise uniqueness of strong solutions by virtue of the Yamada-Watanabe Theorem, and then we investigate the asymptotic properties of invariant measures in two aspects: the first is, focusing on the stochastic systems, to prove the existence of an invariant measure in the case of multiplicative noise, and then prove that the invariant measure can be unique and exponentially mixing if we are restricted to degenerate additive noise. The second is, from the perspective of stochastic stability, to prove that the zero-noise limits of stationary measures for the perturbed stochastic systems are invariant for the deterministic system and their supports are concentrated on the Birkhoff center.

Harnack inequality and gradient estimate for G-SDEs with degenerate noise

In this paper, the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of measure. Moreover, for any bounded and continuous function f, the gradient estimate $$\left| {\nabla {{\bar P}_t}f} \right|\leqslant c(p,t){(\bar P_t{\left| f \right|^p})^{{1 \over p}}},\;\;\;\;\;\;p > 1,\;\;\;\;\;t> 0$$ is obtained for the associated nonlinear semigroup \({\bar P_t}\). As an application of the Harnack inequality, we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition. Finally, an example is presented. All of the above results extend the existing ones in the linear expectation setting.

A general theory of coexistence and extinction for stochastic ecological communities.

We analyze a general theory for coexistence and extinction of ecological communities that are influenced by stochastic temporal environmental fluctuations. The results apply to discrete time (stochastic difference equations), continuous time (stochastic differential equations), compact and non-compact state spaces and degenerate or non-degenerate noise. In addition, we can also include in the dynamics auxiliary variables that model environmental fluctuations, population structure, eco-environmental feedbacks or other internal or external factors. We are able to significantly generalize the recent discrete time results by Benaim and Schreiber (J Math Biol 79:393-431, 2019) to non-compact state spaces, and we provide stronger persistence and extinction results. The continuous time results by Hening and Nguyen (Ann Appl Probab 28(3):1893-1942, 2018a) are strengthened to include degenerate noise and auxiliary variables. Using the general theory, we work out several examples. In discrete time, we classify the dynamics when there are one or two species, and look at the Ricker model, Log-normally distributed offspring models, lottery models, discrete Lotka-Volterra models as well as models of perennial and annual organisms. For the continuous time setting we explore models with a resource variable, stochastic replicator models, and three dimensional Lotka-Volterra models.

Convergence of stochastic-extended Lagrangian molecular dynamics method for polarizable force field simulation

Extended Lagrangian molecular dynamics (XLMD) is a general method for performing molecular dynamics simulations using quantum and classical many-body potentials. Recently several new XLMD schemes have been proposed and tested on several classes of many-body polarization models such as induced dipoles or Drude charges, by creating an auxiliary set of these same degrees of freedom that are reversibly integrated through time. This gives rise to a singularly perturbed Hamiltonian system that provides a good approximation to the time evolution of the real mutual polarization field. To further improve upon the accuracy of the XLMD dynamics in the context of classical polarizable force field simulation, and to potentially extend it to other many-body potentials, we introduce a stochastic modification which leads to a set of singularly perturbed Langevin equations with degenerate noise. We prove that the resulting Stochastic-XLMD converges to the accurate dynamics, and the convergence rate is both sharp and is independent of the accuracy of the initial polarization field. We carefully study the scaling of the damping factor and numerical noise for efficient numerical simulation for Stochastic-XLMD, and we demonstrate the effectiveness of the method for water molecules described by a polarizable force field.

Bismut formula for Lions derivative of distribution-path dependent SDEs

To characterize the regularity of distribution-path dependent SDEs in the initial distribution which varies as probability measure on the path space, we introduce the intrinsic and Lions derivatives for probability measures on Banach spaces, and prove the chain rule of the Lions derivative for the distribution of Banach-valued random variables. By using Malliavin calculus, we establish the Bismut type formula for the Lions derivatives of functional solutions to SDEs with distribution-path dependent drifts. When the noise term is also path dependent so that the Bismut formula is invalid, we establish the asymptotic Bismut formula. Both non-degenerate and degenerate noises are considered. The main results of this paper generalize and improve the corresponding ones derived recently in the literature for the classical SDEs with memory and McKean-Vlasov SDEs without memory.

Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs

<p style='text-indent:20px;'>By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with non-degenerate and degenerate noises.</p>

Numerical computations of geometric ergodicity for stochastic dynamics

A probabilistic approach to compute the geometric convergence rate of a stochastic process is introduced in this paper. The goal is to quantitatively compute both the upper and lower bounds for rate of the exponential convergence to the stationary distribution of a stochastic dynamical system. By applying the coupling method, we derive an algorithm which does not rely on the discretization of the infinitesimal generator. In this way, our approach works well for many high-dimensional examples. We apply this algorithm to the random perturbations of both iterative maps and differential equations. We show that the rate of geometric ergodicity of a random perturbed system can, to some extent, reveal the degree of chaoticity of the underlying deterministic dynamics. Various SDE models including the ones with degenerate noise or living on the high-dimensional state space are also explored.

Towards mesoscopic ergodic theory

The present paper is devoted to a preliminary study towards the establishment of an ergodic theory for stochastic differential equations (SDEs) with less regular coefficients and degenerate noises. These equations are often derived as mesoscopic limits of complex or huge microscopic systems. By studying the associated Fokker-Planck equation (FPE), we prove the convergence of the time average of globally defined weak solutions of such an SDE to the set of stationary measures of the FPE under Lyapunov conditions. In the case where the set of stationary measures consists of a single element, the unique stationary measure is shown to be physical. Similar convergence results for the solutions of the FPE are established as well. Some of our convergence results, while being special cases of those contained in Ji et al. (2019) for SDEs with periodic coefficients, have weaken the required Lyapunov conditions and are of much simplified proofs. Applications to stochastic damping Hamiltonian systems and stochastic slow-fast systems are given.

Asymptotic Log-Harnack Inequality and Ergodicity for 3D Leray-α Model with Degenerate Type Noise

The asymptotic log-Harnack inequality is proved for Leray-α model with degenerate type noise using the asymptotic coupling method. In particular, we don’t impose any lower bound assumption for the viscosity constant. As applications, we also derive ergodicity and further asymptotic properties for stochastic 3D Leray-α model.

Exponential mixing properties of the stochastic tamed 3D Navier–Stokes equation with degenerate noise

The current paper is devoted to the stochastic tamed 3D Naiver-Stokes equation with degenerate noise with degenerate random forcing. We prove that the associated Markov semigroup possesses an exponentially attracting invariant measure by showing that the corresponding dynamical system possesses a strong type of Lyapunov structure along with a special type of gradient inequality and is of a relatively weak form of irreducibility.

Asymptotic log-Harnack inequality and applications for stochastic 2D hydrodynamical-type systems with degenerate noise

In this paper, an asymptotic log-Harnack inequality and some consequent properties are established via the asymptotic coupling method for a class of stochastic 2D hydrodynamical-type systems driven by degenerate noise. The main results are applicable to the stochastic 2D Navier–Stokes equations, stochastic 2D magneto-hydrodynamic equations and stochastic 2D Boussinesq equations, stochastic 2D magnetic Benard problem, stochastic 3D Leray- $$\alpha $$ model and also stochastic shell models of turbulence in the degenerate noise case.