Application of group analysis to stochastic equations of fluid dynamics

Group analysis is used to study stochastic equations of fluid dynamics. Determining equations for admitted Lie groups of transformation involving independent and dependent variables and Wiener processes are obtained. It is shown that, as in the case of deterministic differential equations, admitted groups make it possible to reduce invariant solutions of stochastic differential equations to solutions with a smaller number of independent variables.     

Stationary Solutions for Stochastic Differential Equations Driven by Lévy Processes

In the paper, stationary solutions of stochastic differential equations driven by Lévy processes are considered. And the existence of these stationary solutions follows from the theory of random dynamical systems and their attractors. Moreover, under a one-sided Lipschitz continuity condition and a temperedness condition, Itô and Marcus stochastic differential equations driven by Lévy processes are proved to have stationary solutions. Besides, continuous dependence of stationary solutions on drift coefficients of these equations is presented.     

ON SOLUTIONS OF BACKWARD STOCHASTIC DIFFERENTIAL EQUATIONS WITH JUMPS, WITH UNBOUNDED STOPPING TIMES AS TERMINAL AND WITH NON-LIPSCHITZ COEFFICIENTS, AND PROBABILISTIC INTERPRETATION OF QUASI-LINEAR ELLIPTIC TYPE INTEGRO- DIFFERENTIAL EQUATIO

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Stochastic differential equations for the kinetic and hydrodynamic equations

A finite system of stochastic interacting particles is considered. The system approximates the solutions of the kinetic equations (the Boltzmann equation, the Boltzmann-Enskog equation) as well as the solutions describing the macroscopic evolution of fluids: the Euler and the Navier-Stokes hydrodynamic equations.     

平面弹性力学充要的随机边界元

将平面弹性力学确定性的充分必要的边界积分方程推广到含材料常数随机的不确定问题中去，给出了位移的均值以及偏差的充分必要的边界积分方程。数值计算结果表明，和确定性的积分方程一样，习用的随机边界积分方程在退化尺度附近，无论是均值还是偏差都存在巨大的误差，而充要的随机边界积分方程则始终保持良好的精度     

广义密度演化方程的δ函数序列解法

随机动力系统响应或状态向量的概率密度函数一般遵循概率密度演化方程,如Liouville方程、FPK方程和Dostupov-Pugachev方程,但是上述方程均属于高维偏微分方程,求解相当困难. 基于概率守恒原理的随机事件描述导出的广义密度演化方程,其维数与系统自由度无关,为随机动力系统分析提供了可能的途径. 从广义密度演化方程的形式解出发,引入δ函数的渐近序列,获得了广义密度演化方程的一种新的数值解法------广义密度演化方程的δ序列解法. 将建议方法与非参数密度估计进行了对比,指出非参数密度估计是该方法的一个特例. 最后,分别采用重构实例和演化实例验证了该方法在一维和多维情形下的有效性.      

Non-linear and linear evolution of perturbation in stochastic basic flows

In this paper, we examine the non-linear and linear evolutions of perturbation in stochastic basic flows with two-dimensional quasi-geostrophic equations on a sphere. As the analytic solutions for the considered quasi-geostrophic equations are not available, the Fourier finite volume element method is used to perform numerical simulation. It is found that, the non-linear and linear evolutions of perturbation in stochastic basic flow will be consistent for a short period of time and small stochastic fluctuations when they are consistent in the deterministic basic flow. However, the tangent linear model will fail to approximate the original non-linear model when the time period is considerably long and stochastic fluctuation becomes large. Moreover, the global energy decays faster for stochastic basic flow with stronger fluctuations.     

Exact linearization of one dimensional Itô equations driven by fBm: Analytical and numerical solutions

Necessary and sufficient conditions for the linearization of the one-dimensional Itô stochastic differential equations driven by fractional Brownian motion (fBm) are given. Stochastic integrating factor has been introduced. A modified Milstein method has been developed to obtain numerical solutions. Analytical solutions have been compared with the numerical solutions for linearizable equations.     

On solutions of backward stochastic differential equations with jumps, with unbounded stopping times as terminal and with non-lipschitz coefficients, and probabilistic interpretation of quasi-linear elliptic type integro-differential equations

The existence and uniqueness of solutions to backward stochastic differential equations with jumps and with unbounded stopping time as terminal under the non-Lipschitz condition are obtained. The convergence of solutions and the continuous dependence of solutions on parameters are also derived. Then the probabilistic interpretation of solutions to some kinds of quasi-linear elliptic type integro-differential equations is obtained. Foundation item: the National Natural Science Foundation of China (79790130); the Foundation of Zhongshan University Front Research Biography: Situ Rong (1935 −)     

LINEAR QUADRATIC NONZERO-SUM DIFFERENTIAL GAMES WITH RANDOM JUMPS

IntroductionFully coupled forward-backward stochastic differential equations with Brownian motioncan be encountered in the optimization problem when we apply stochastic maximum principleand in mathematics finance when we consider large investor in securit…     

LINEAR QUADRATIC NONZERO-SUM DIFFERENTIAL GAMES WITH RANDOM JUMP

The existence and uniqueness of the solutions for one kind of forward-backward stochastic differential equations with Brownian motion and Poisson process as the noise source were given under the monotone conditions.Then these results were applied to nonzero-sum differential games with random jumps to get the explicit form of the open-loop Nash equilibrium point by the solution of the forward-backward stochastic differential equations.     

Nonlinear Fluctuations of Weakly Asymmetric Interacting Particle Systems

We introduce what we call the second-order Boltzmann–Gibbs principle, which allows one to replace local functionals of a conservative, one-dimensional stochastic process by a possibly nonlinear function of the conserved quantity. This replacement opens the way to obtain nonlinear stochastic evolutions as the limit of the fluctuations of the conserved quantity around stationary states. As an application of this second-order Boltzmann–Gibbs principle, we introduce the notion of energy solutions of the KPZ and stochastic Burgers equations. Under minimal assumptions, we prove that the density fluctuations of one-dimensional, stationary, weakly asymmetric, conservative particle systems are sequentially compact and that any limit point is given by energy solutions of the stochastic Burgers equation. We also show that the fluctuations of the height function associated to these models are given by energy solutions of the KPZ equation in this sense. Unfortunately, we lack a uniqueness result for these energy solutions. We conjecture these solutions to be unique, and we show some regularity results for energy solutions of the KPZ/Burgers equation, supporting this conjecture.     

First-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations

In this paper, first-passage problem of a class of internally resonant quasi-integrable Hamiltonian system under wide-band stochastic excitations is studied theoretically. By using stochastic averaging method, the equations of motion of the original internally resonant Hamiltonian system are reduced to a set of averaged Itô stochastic differential equations. The backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the mean first-passage time are established under appropriate boundary and (or) initial conditions. An example is given to show the accuracy of the theoretical method. Numerical solutions of high-dimensional backward Kolmogorov and Pontryagin equation are obtained by finite difference. All theoretical results are verified by Monte Carlo simulation.     

悬臂轴拉杆一阶随机分析的解析解

针对悬臂轴拉杆在自由端受集中力问题进行了随机分析，其中考虑了轴拉杆杨氏模量的变异，在小变异情况下获得了不同相关结构与相关长度下问题的解析解，并就结果对相关结构的敏感性进行了讨论．     

Random Attractors for Degenerate Stochastic Partial Differential Equations

We prove the existence of random attractors for a large class of degenerate stochastic partial differential equations (SPDE) perturbed by joint additive Wiener noise and real, linear multiplicative Brownian noise, assuming only the standard assumptions of the variational approach to SPDE with compact embeddings in the associated Gelfand triple. This allows spatially much rougher noise than in known results. The approach is based on a construction of strictly stationary solutions to related strongly monotone SPDE. Applications include stochastic generalized porous media equations, stochastic generalized degenerate $p$ -Laplace equations and stochastic reaction diffusion equations. For perturbed, degenerate $p$ -Laplace equations we prove that the deterministic, $\infty $ -dimensional attractor collapses to a single random point if enough noise is added.     

Existence and Uniqueness for Stochastic 2D Euler Flows with Bounded Vorticity

The strong existence and the pathwise uniqueness of solutions with \({L^{\infty}}\)-vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved.     

Stochastic averaging of energy harvesting systems

A stochastic averaging method is proposed for nonlinear energy harvesters subjected to external white Gaussian noise and parametric excitations. The Fokker–Planck–Kolmogorov equation of the coupled electromechanical system of energy harvesting is a three variables nonlinear parabolic partial differential equation whose exact stationary solutions are generally hard to find. In order to overcome difficulties in solving higher dimensional nonlinear partial differential equations, a transformation scheme is applied to decouple the electromechanical equations. The averaged Itô equations are derived via the standard stochastic averaging method, then the FPK equations of the decoupled system are obtained. The exact stationary solution of the averaged FPK equation is used to determine the probability densities of the displacement, the velocity, the amplitude, the joint probability densities of the displacement and velocity, and the power of the stationary response. The effects of the system parameters on the output power are examined. The approximate analytical outcomes are qualitatively and quantitatively supported by the Monte Carlo simulations.     

Global Attractors for 3-Dimensional Stochastic Navier–Stokes Equations

Sell's approach 35 to the construction of attractors for the Navier-Stokes equations in 3-dimensions is extended to the 3D stochastic equations with a general multiplicative noise. The new notion of a process attractor is defined as a set A of processes, living on a single filtered probability space, that is a set of solutions and attracts all solution processes in a given class. This requires the richness of a Loeb probability space. Non-compactness results for A and a characterization in terms of two-sided solutions are given.     

First passage failure of quasi-partial integrable generalized Hamiltonian systems

The first passage failure of quasi-partial integrable generalized Hamiltonian systems is studied by using the stochastic averaging method. First, the stochastic averaging method for quasi-partial integrable generalized Hamiltonian systems is introduced briefly. Then, the backward Kolmogorov equation governing the conditional reliability function and the Pontryagin equation governing the conditional mean of first passage time are derived from the averaged Itô equations. The conditional reliability function, the conditional probability density and mean of the first passage time are obtained from solving these equations together with suitable initial condition and boundary conditions, respectively. Finally, one example is given to illustrate the proposed procedure in detail and the solutions are confirmed by using the results from Monte Carlo simulation of the original system.     

Stochastic response of an axially loaded composite Timoshenko beam exhibiting bending–torsion coupling

A spectral finite element method is proposed to investigate the stochastic response of an axially loaded composite Timoshenko beam with solid or thin-walled closed section exhibiting bending–torsion materially coupling under the stochastic excitations with stationary and ergodic properties. The effects of axial force, shear deformation (SD) and rotary inertia (RI) as well as bending–torsion coupling are considered in the present study. First, the damped general governing differential equations of motion of an axially loaded composite Timoshenko beam are derived. Then, the spectral finite element formulation is developed in the frequency domain using the dynamic shape functions based on the exact solutions of the governing equations in undamped free vibration, which is used to compute the mean square displacement response of axially loaded composite Timoshenko beams. Finally, the proposed method is illustrated by its application to a specific example to investigate the effects of bending–torsion coupling, axial force, SD and RI on the stochastic response of the composite beam.