Stochastic Heat Equation with Multiplicative Fractional-Colored Noise

We consider the stochastic heat equation with multiplicative noise \(u_{t}=\frac{1}{2}\Delta u+u\dot{W}\) in ?₊×? ^d , whose solution is interpreted in the mild sense. The noise \(\dot{W}\) is fractional in time (with Hurst index H≥1/2), and colored in space (with spatial covariance kernel f). When H>1/2, the equation generalizes the Itô-sense equation for H=1/2. We prove that if f is the Riesz kernel of order α, or the Bessel kernel of order α<d, then the sufficient condition for the existence of the solution is d≤2+α (if H>1/2), respectively d<2+α (if H=1/2), whereas if f is the heat kernel or the Poisson kernel, then the equation has a solution for any d. We give a representation of the kth order moment of the solution in terms of an exponential moment of the “convoluted weighted” intersection local time of k independent d-dimensional Brownian motions.     

广义Kuramoto-Sivashinsky方程强解的全局吸引子的存在性

Using a new method developed in[5],we prove the existence of global attractors for the Generalized Kuramoto-Sivashinsky equation in H3per(Ω)and H4per(Ω).     

Random Attractors for the Stochastic Kuramoto-Sivashinsky Equation

Abstract

Random systems may be more reasonable by incorporating influence of noise into deterministic systems. The notion of a random attractor is one of the very basic concepts of the theory of random dynamical systems. In this article, we consider the well-known Kuramoto–Sivashinsky equation with stochastic perturbation. Our aim is to attempt to obtain a so-called pull-back random attractor for stochastic Kuramoto–Sivashinsky equation. In particular, the Hausdorff dimension of a random attractor is finite. For simplicity, we always restrict ourselves to odd initial conditions, but the result for all initial conditions is also true.     

The Stochastic Wave Equation with Multiplicative Fractional Noise: A Malliavin Calculus Approach

We consider the stochastic wave equation with multiplicative noise, which is fractional in time with index H > 1/2, and has a homogeneous spatial covariance structure given by the Riesz kernel of order α. The solution is interpreted using the Skorohod integral. We show that the sufficient condition for the existence of the solution is α > d − 2, which coincides with the condition obtained in Dalang (Electr J Probab 4(6):29, 1999), when the noise is white in time. Under this condition, we obtain estimates for the p-th moments of the solution, we deduce its H?lder continuity, and we show that the solution is Malliavin differentiable of any order. When d ≤ 2, we prove that the first-order Malliavin derivative of the solution satisfies a certain integral equation.     

The Spectral Method for the Generalized Kuramoto-Sivashinsky Equation

A spectral method is proposed, the existence and uniqueness of the global and smooth solution are proved for the periodic initial value problem of the generalized K-S equation. The error estimates are established and the convergence is proved for the approximate solution of the spectral method.     

具有线性乘积白噪声的随机非自治吊桥方程的随机指数吸引子

本文考虑具有线性乘积白噪声的随机非自治吊桥方程长时间行为.首先,建立了所研究共圈系统的适定性;第二步,研究了该系统随机吸引子的存在性;第三步,当随机系数趋于0时,得到了随机吸引子的上半连续性;第四步,通过``迭代''法证明了随机吸引子在高正则空间中的正则性;最后,给出了该系统随机指数吸引子的存在性,同时得到了吸引子的有限分形维数.     

带加性白噪声的随机广义Boussinesq方程解的爆破性

本文研究一维空间中带加性白噪声的随机广义Boussinesq方程.首先运用截断方法,建立该方程Canchy问题解的局部适定性,然后运用It(o)公式导出了该系统的能量方程,最后证明了该系统解的爆破性.     

一类广义长短波方程的整体适定性

In the present paper,we investigate the well-posedness of the global solution for the Cauchy problem of generalized long-short wave equations.Applying Kato's method for abstract quasi-linear evolution equations and a priori estimates of solution,we get the existence of globally smooth solution.     

Global Well-posedness and Global Attractor for Two-dimensional Zakharov-Kuznetsov Equation

The initial value problem for two-dimensional Zakharov-Kuznetsov equation is shown to be globally well-posed in H^s(R²) for all 5/7 < s < 1 via using I-method in the context of atomic spaces. By means of the increment of modified energy, the existence of global attractor for the weakly damped, forced Zakharov-Kuznetsov equation is also established in H^s(R²) for 10/11 < s < 1.     

The Asymptotic Behavior of the Stochastic Nonlinear Schr(o)dinger Equation With Multiplicative Noise

The nonlinear Schr(o)dinger equation is one of the basic models for nonlinear waves. In some circumstances, randomness has to be taken into account and it often occurs through a random potential. Here, we consider the following equation     

带附加噪声的随机广义2D Ginzburg-Landau方程的渐进行为

考虑带附加噪声的随机广义2D Ginzburg-Landau方程.通过先验估计的方法,随机动力系统的紧性得到证明,进一步验证了该随机动力系统在础存在随机整体吸引子.     

Well-posedness and Large Deviations of the Stochastic Modified Camassa-Holm Equation

We derive the heat trace asymptotics of the generator of subordinate Brownian motion on Euclidean space for a class of Laplace exponents. The terms in the asymptotic expansion can be computed to arbitrary order and depend both on the geometry of Euclidean space and the short-time behaviour of the process. If the Blumenthal-Getoor index of the process is rational, then the asymptotics may contain logarithmic terms. The key assumption is the existence of a suitable density for the Lévy measure of the subordinator. The analysis is highly explicit.     

非局部Kuramoto-Sivashinsky方程整体吸引子的维数估计

本文主要讨论NK-S方程整体吸引子的Hausdorff维数的上界估计.     

Stochastic Equations with an Unbounded Operator Coefficient and Multiplicative Noise

Under study is a stochastic operator-differential equation with multiplicative noise in the space of Hilbert-valued generalized random variables. Existence and uniqueness of solutions to the Cauchy problem are proved for the case of an unbounded operator coefficient at the white noise. The equation of population dynamics with a stochastically perturbed multiplication operator is presented as an example.     

Large Deviations for Stochastic Evolution Equations with Small Multiplicative Noise

The Freidlin-Wentzell large deviation principle is established for the distributions of stochastic evolution equations with general monotone drift and small multiplicative noise. As examples, the main results are applied to derive the large deviation principle for different types of SPDE such as stochastic reaction-diffusion equations, stochastic porous media equations and fast diffusion equations, and the stochastic p-Laplace equation in Hilbert space. The weak convergence approach is employed in the proof to establish the Laplace principle, which is equivalent to the large deviation principle in our framework.     

Stochastic Flow and Noise Associated with the Tanaka Stochastic Differential Equation

We study the properties of the noise (in the Tsirelson sense) that is generated by the solutions of the well-known Tanaka equation.     

随机广义KdV方程的随机精确解

通过一个辅助性方程和埃米尔特变换研究广义随机KdV方程的随机雅克比椭圆函数类波解.此外,还通过椭圆函数在模数取极限m→0和m→1的情况,给出方程的随机类孤子解和随机三角函数波解,所得结果丰富了广义随机KdV方程的精确解.     

Global Maximization of a Generalized Concave Multiplicative Function

This article presents a branch-and-bound algorithm for globally solving the problem (P) of maximizing a generalized concave multiplicative function over a compact convex set. Since problem (P) does not seem to have been studied previously, the algorithm is apparently the first algorithm to be proposed for solving this problem. It works by globally solving a problem (P1) equivalent to problem (P). The branch-and-bound search undertaken by the algorithm uses rectangular partitioning and takes place in a space which typically has a much smaller dimension than the space to which the decision variables of problem (P) belong. Convergence of the algorithm is shown; computational considerations and benefits for users of the algorithm are given. A sample problem is also solved.     

广义旋转Navier-Stokes方程解的整体适定性和解析性

在α和q满足适当的条件下,当初值属于Fourier-Herz空间?_q^1-2α（R³）时,我们建立了广义3维不可压旋转Navier-Stokes方程温和解的整体适定性和解析性.作为推论,我们也给出了广义Navier-Stokes方程的相应结论.     

环状区域中的轴对称Kuramoto-Sivashinsky方程的整体吸引子

在本文中，环状区域中的轴对称Ｋｕｒａｍｏｔｏ－Ｓｉｖａｓｈｉｎｓｋｙ方程的有限维整体吸引子被得到了．