On stochastically perturbed differential systems

We give a result which refers to the global existence and pathwise uniqueness of solutions of Itô stochastic differential systems. The sufficient conditions we propose improve some results of Taniguchi [5] and Constantin [1].     

A uniformly exponential random forward attractor which is not a pullback attractor

A compact random set is constructed which is invariant with respect to a random dynamical system, and which is an exponential forward attractor, but fails to be a pullback attractor.     

On weak convergence of solutions of random perturbed evolution equations

We consider the weak convergence of measures generated by solutions of linear evolution equations depending on diffusion processes to the Gaussian measure as a small parameter tends to zero.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 2, pp. 213–219, February, 1995.     

Parameter estimation for the stochastically perturbed Navier-Stokes equations

We consider a parameter estimation problem of determining the viscosity ν of a stochastically perturbed 2D Navier-Stokes system. We derive several different classes of estimators based on the first N Fourier modes of a single sample path observed on a finite time interval. We study the consistency and asymptotic normality of these estimators. Our analysis treats strong, pathwise solutions for both the periodic and bounded domain cases in the presence of an additive white (in time) noise.     

A random attractor for a stochastic second order lattice system with random coupled coefficients

In this paper, we mainly focus on the asymptotic behavior of solutions to the second-order stochastic lattice equations with random coupled coefficients and multiplicative white noises in weighted spaces of infinite sequences. We first transfer stochastic lattice equations into random lattice equations and prove the existence and uniqueness of solutions which generate a random dynamical system. Second we consider the existence of a tempered random bounded absorbing set and a random attractor for the system. Then we establish the upper semicontinuity of random attractors as the coefficient of the white noise term tends to zero. Finally we present the corresponding results for the system with additive white noises.     

Averaging principle for perturbed random evolution equations and corresponding Dirichlet problems

Summary We continue the study of exit problems for perturbed random evolution equations corresponding to the weakly coupled elliptic PDE systems. In the present paper we consider the cases where the corresponding random evolutions stay in a given domain for ever with probability one, but do not hinder the exit of the perturbed process. We treat such problems by methods based on the averaging principle. In such a way we also study the asymptotic behavior of the solutions of the corresponding perturbed Dirichlet problems.Supported in part by US-Israel BSFSponsored in part by the Landau Center for Mathematical Research in Analysis supported by the Minerva Foundation (Federal Republic of Germany)     

On the dimension of the attractor for a perturbed 3d Ladyzhenskaya model

We consider the so-called Ladyzhenskaya model of incompressible fluid, with an additional artificial smoothing term ?Δ³. We establish the global existence, uniqueness, and regularity of solutions. Finally, we show that there exists an exponential attractor, whose dimension we estimate in terms of the relevant physical quantities, independently of ? > 0.     

Snakes and perturbed random walks

We study some properties of random walks perturbed at extrema, which are generalizations of the walks considered, e.g., by Davis (1999) and Tóth (1996). This process can also be viewed as a version of an excited random walk, recently studied by many authors. We obtain several properties related to the range of the process with infinite memory and prove the strong law, the central limit theorem, and the criterion for the recurrence of the perturbed walk with finite memory. We also state some open problems. Our methods are predominantly combinatorial and do not involve complicated analytic techniques.     

Effect of bounded noise on chaotic motions of stochastically perturbed slowly varying oscillator

The effect of bounded noise on the chaotic behavior of a class of slowly varying oscillators is investigated. The stochastic Melnikov method is employed and then the criteria in both mean and mean-square sense are derived. The threshold amplitude of bounded noise given by stochastic Melnikov process is in good comparison with one determined by the numerical simulation of top Lyapunov exponents. The presence of noise scatters the chaotic domain in parameter space and the larger noise intensity results in a sparser and more irregular region. Both the simple cell mapping method and the generalized cell mapping method are applied to demonstrate the effects of noises on the attractors. Results show that the attractors are diffused and smeared by bounded noise and if the noise intensity increases, the diffusion is exacerbated.     

Second-order general perturbed sweeping process differential inclusion

The paper presents a study of perturbed sweeping process where the moving set depends on both the time and the state. This evolution problem is governed by second-order differential inclusions with an unbounded perturbation. Assuming that such set is prox-regular or subsmooth, we prove the existence of solutions even in the presence of a delay.     

An attractor for the singularly perturbed Kirchhoff equation with supercritical nonlinearity

We consider a quasilinear wave equation of Kirchhoff type

?_utt−(1+‖∇u‖²)Δu+_ut+f(u)=g(x),$?u_{tt}-(1+{\Vert \mathrm{\nabla }u\Vert }^2)\mathrm{\Delta }u+u_t+f(u)=g(x),$

where ?>0$?>0$ is a small parameter. Without any growth restrictions on the nonlinearity f(u)$f(u)$, we prove the existence of a finite-dimensional global attractor on an appropriate (bounded) phase space. The key step is the estimate of the difference between the solutions of a quasilinear dissipative hyperbolic equation of Kirchhoff type and the corresponding quasilinear parabolic equation.     

Almost sure convergence for stochastically biased random walks on trees

We are interested in the biased random walk on a supercritical Galton?CWatson tree in the sense of Lyons (Ann. Probab. 18:931?C958, 1990) and Lyons, Pemantle and Peres (Probab. Theory Relat. Fields 106:249?C264, 1996), and study a phenomenon of slow movement. In order to observe such a slow movement, the bias needs to be random; the resulting random walk is then a tree-valued random walk in random environment. We investigate the recurrent case, and prove, under suitable general integrability assumptions, that upon the system??s non-extinction, the maximal displacement of the walk in the first n steps, divided by (log n)³, converges almost surely to a known positive constant.