Lyapunov exponents and stationary solutions for affine stochastic delay equations

A certain class of affine delay equations is considered. Two cases for the forcingfunction M are treated: M locally integrable deterministic, and M a random process with stationaryincrements. The Lyapunov spectrum of the homogeneous equation is used to decompose the state spaceinto finite-dimensional and finite-codimensional subspaces. Using a suitable variation of constants representation, formulas for the projection of the trajectories onto the above subspaces are obtained. If the homogeneous equation is hyperbolic and M has stationary increments, existence and uniqueness of a stationary solution for the affine stochastic delay equation is proved. The existence of Lyapunov exponents for the affine equation and their dependence on initial conditions is als studied.     

Stationary Conjugation of Flows for Parabolic SPDEs with Multiplicative Noise and Some Applications

Abstract

The purpose of this paper is to transform a nonlinear stochastic partial differential equation of parabolic type with multiplicative noise into a random partial differential equation by using a bijective random process. A stationary conjugation is constructed, which is of interest for asymptotic problems. The conjugation is used here to prove the existence of the stochastic flow, the perfect cocycle property and the existence of the random attractor, all nontrivial properties in the case of multiplicative noise.     

Solitary waves for Korteweg-de Vries equation with small delay

This paper deals with the existence of solitary waves for Korteweg-de Vries equation with time delay. Based upon the inertial manifold theory and differential manifold geometric theory, the existence of solitary wave solution is proved when the delay is small enough. Up to now, studies on solitary wave for such delay differential equation are not available, so the results of this paper are new.     

Construction of an Inertial Manifold of a Parabolic Equation with a Monotonic Nonlinear Part

Sufficient conditions are obtained for the existence of an inertial manifold of a parabolic equation with a monotonic nonlinear part in a bounded region of not more than three dimensions. The existence of an inertial manifold is proved for a reaction diffusion equation viewed on a two-dimensional torus.     

Weak KAM Theory topics in the stationary ergodic setting

We perform a qualitative analysis of the critical equation associated with a stationary ergodic Hamiltonian through a stochastic version of the metric method, where the notion of closed random stationary set, issued from stochastic geometry, plays a major role. Our purpose is to give an appropriate notion of random Aubry set, to single out characterizing conditions for the existence of exact or approximate correctors, and write down representation formulae for them. For the last task, we make use of a Lax-type formula, adapted to the stochastic environment. This material can be regarded as a first step of a long-term project to develop a random analog of Weak KAM Theory, generalizing what done in the periodic case or, more generally, when the underlying space is a compact manifold.     

Random attractor of stochastic complex Ginzburg–Landau equation with multiplicative noise on unbounded domain

The existence of a compact random attractor for the stochastic complex Ginzburg–Landau equation with multiplicative noise has been investigated on unbounded domain. The solutions are considered in suitable spaces with weights. According to crucial properties of Ornstein–Uhlenbeck process, using the tail-estimates method, the key uniform a priori estimates for the tail of solutions have been obtained, which give the asymptotic compactness of random attractors. Then the existence of a compact random attractor for the corresponding dynamical system is proved in suitable spaces with weights.     

Dynamical Analysis for a General Jerky Equation with Random Excitation

A general jerky equation with random excitation is investigated in this paper. Before introducing the random excitation term, the equation is reduced to a two-dimensional model when undergoing a Hopf bifurcation. Then the model with the parametric excitation and external excitation is converted to a stochastic differential equation with singularity based on the stochastic average theory. For the equation, its dynamical behaviors are analyzed in different parameters'' spaces, including the stability, stochastic bifurcation and stationary solution. Besides, numerical simulations are given to show the asymptotic behavior of the stationary solution.     

Stability of Riccati's Equation with Random Stationary Coefficients

The purpose of this paper is to study under weak conditions of stabilizability and detectability, the asymptotic behavior of the matrix Riccati equation which arises in stochastic control and filtering with random stationary coefficients. We prove the existence of a stationary solution of this Riccati equation. This solution is attracting, in the sense that if P _t is another solution, then onverges to 0 exponentially fast as t tends to +∞ , at a rate given by the smallest positive Lyapunov exponent of the associated Hamiltonian matrices. Accepted 13 January 1998     

Backward Stochastic Riccati Equations and Infinite Horizon L-Q Optimal Control with Infinite Dimensional State Space and Random Coefficients

We study the Riccati equation arising in a class of quadratic optimal control problems with infinite dimensional stochastic differential state equation and infinite horizon cost functional. We allow the coefficients, both in the state equation and in the cost, to be random. In such a context backward stochastic Riccati equations are backward stochastic differential equations in the whole positive real axis that involve quadratic non-linearities and take values in a non-Hilbertian space. We prove existence of a minimal non-negative solution and, under additional assumptions, its uniqueness. We show that such a solution allows to perform the synthesis of the optimal control and investigate its attractivity properties. Finally the case where the coefficients are stationary is addressed and an example concerning a controlled wave equation in random media is proposed.     

Global attractors and approximate inertial

An abstract nonautonomous differential equation, u' + Au + F(u) = f ( t ) , is considered using assumptions appropriate for systems of reaction-diffusion equations on multi-dimensional spatial domains. A priori estimates establish the existence of absorbing balls in relevant function spaces, and nonsequently the existence of a global attractor is verified in the associated skew-product flow. It is also shown that approximate inertial manifolds exist for this equation. These are finite-dimensional manifolds which have exponentially attracting neighborhoods under the flow. The dimension of the manifold and the thickness of the attracting neighborhood are inversely related.     

Girsanov's theorem and ergodic properties of statistical solutions of nonlinear parabolic equations

Theorems on the existence and uniqueness of the statistical solutions and for the existence of stationary solutions for nonlinear stochastic equations are proved. Markov families of statistical solutions are constructed. The ergodic properties of monotone systems are investigated. A Girsanov type theorem on the absolute continuity of the statistical solution of a nonlinear stochastic parabolic equation with respect to the statistical solution of the monotone equation is proved.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 12, pp. 88–117, 1987.     

Inertial manifolds and linear multi-step methods

We determine the existence and C ¹ convergence of an inertial manifold for a strongly A(α) stable, pth order, p≧1, linear multi-step method approximating a sectorial evolution equation that satisfies a gap condition. This inertial manifold gives rise to a one-step method that C ¹ approximates the inertial form of the evolution equation and yields further approximation properties of the multi-step method. This revised version was published online in June 2006 with corrections to the Cover Date.     

Wong-Zakai approximations and attractors for fractional stochastic reaction-diffusion equations on unbounded domains

In this paper, we investigate the Wong-Zakai approximations induced by a stationary process and the long term behavior of the fractional stochastic reaction-diffusion equation driven by a white noise. Precisely, one of the main ingredients in this paper is to establish the existence and uniqueness of tempered pullback attractors for the Wong-Zakai approximations of fractional stochastic reaction-diffusion equations. Thereafter the upper semi-continuity of attractors for the Wong-Zakai approximation of the equation as $\delta\rightarrow0$ is proved.     

一类非自治分数阶随机波动方程的随机吸引子

本文考虑带加性噪声的非自治分数阶随机波动方程在无界区域R~n上的渐近行为.首先将随机偏微分方程转化为随机方程,其解产生一个随机动力系统,然后运用分解技术建立该系统的渐近紧性,最后证明随机吸引子的存在性.     

The Burgers equation with affine linear noise: Dynamics and stability

We study the dynamics of the Burgers equation on the unit interval driven by affine linear noise. Mild solutions of the Burgers stochastic partial differential equation generate a smooth perfect and locally compacting cocycle on the energy space. Using multiplicative ergodic theory techniques, we establish the existence of a discrete non-random Lyapunov spectrum for the cocycle. We establish a local stable manifold theorem near a hyperbolic stationary point, as well as the existence of local smooth invariant manifolds with finite codimension and a countable global invariant foliation of the energy space relative to an ergodic stationary point.     

Asymptotic behavior of stochastic Schrödinger lattice systems driven by nonlinear noise

Abstract

We study the random dynamics of the N-dimensional stochastic Schrödinger lattice systems with locally Lipschitz diffusion terms driven by locally Lipschitz nonlinear noise. We first prove the existence and uniqueness of solutions and define a mean random dynamical system associated with the solution operators. We then establish the existence and uniqueness of weak pullback random attractors in a Bochner space. We finally prove the existence of invariant measures of the stochastic equation in the space of complex-valued square-summable sequences. The tightness of a family of probability distributions of solutions is derived by the uniform estimates on the tails of the solutions at far field.     

Weak solution for stochastic differential equations with terminal conditions

The notion of weak solution for stochastic differential equation with terminal conditions is introduced. By Girsanov transformation, the equivalence of existence of weak solutions for two-type equations is established. Several sufficient conditions for the existence of the weak solutions for stochastic differential equation with terminal conditions are obtained, and the solution existence condition for this type of equations is relaxed. Finally, an example is given to show that the result is an essential extension of the one under Lipschitz condition ong with respect to (Y,Z).     

STATIONARY SOLUTION FOR A STOCHASTIC LIENARD EQUATION WITH MARKOVIAN SWITCHING

This paper considers a stochastic Liénard equation with Markovian switching. The Feller continuity of its solution is proved by the coupling method and a truncation argument. The existence of a stationary solution for the equation is also proved under the Foster-Lyapunov drift condition.     

Quaternionic Monge-Ampère equation and Calabi problem for HKT-manifolds

A quaternionic version of the Calabi problem on the Monge-Ampère equation is introduced, namely a quaternionic Monge-Ampère equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For a hypercomplex manifold with holonomy in SL(n,ℍ), uniqueness (up to a constant) of a solution is proven, aas well as the zero order a priori estimate. The existence of a solution is conjectured, similar to the Calabi-Yau theorem. We reformulate this quaternionic equation as a special case of the complex Hessian equation, making sense on any complex manifold.