Ergodicity for a class of semilinear stochastic partial differential equations

In this paper, we establish the existence and uniqueness of invariant measures for a class of semilinear stochastic partial differential equations driven by multiplicative noise on a bounded domain. The main results can be applied to SPDEs of various types such as the stochastic Burgers equation and the reaction-diffusion equations perturbed by space-time white noise.     

Convergence of a random walk method for a partial differential equation

A Cauchy problem for a one-dimensional diffusion-reaction equation is solved on a grid by a random walk method, in which the diffusion part is solved by random walk of particles, and the (nonlinear) reaction part is solved via Euler's polygonal arc method. Unlike in the literature, we do not assume monotonicity for the initial condition. It is proved that the algorithm converges and the rate of convergence is of order , where is the spatial mesh length.

     

Extremes of Gaussian processes with a smooth random variance

Let ξ(t) be a standard stationary Gaussian process with covariance function r(t), and η(t), another smooth random process. We consider the probabilities of exceedances of ξ(t)η(t) above a high level u occurring in an interval [0,T] with T>0. We present asymptotically exact results for the probability of such events under certain smoothness conditions of this process ξ(t)η(t), which is called the random variance process. We derive also a large deviation result for a general class of conditional Gaussian processes X(t) given a random element Y.     

Effects of random noise in a dynamical model of love

This paper aims to investigate the stochastic model of love and the effects of random noise. We first revisit the deterministic model of love and some basic properties are presented such as: symmetry, dissipation, fixed points (equilibrium), chaotic behaviors and chaotic attractors. Then we construct a stochastic love-triangle model with parametric random excitation due to the complexity and unpredictability of the psychological system, where the randomness is modeled as the standard Gaussian noise. Stochastic dynamics under different three cases of “Romeo’s romantic style”, are examined and two kinds of bifurcations versus the noise intensity parameter are observed by the criteria of changes of top Lyapunov exponent and shape of stationary probability density function (PDF) respectively. The phase portraits and time history are carried out to verify the proposed results, and the good agreement can be found. And also the dual roles of the random noise, namely suppressing and inducing chaos are revealed.     

Asymptotic analysis of a kernel estimator for parabolic stochastic partial differential equations driven by fractional noises

We study a strongly elliptic partial differential operator with time-varying coeffcient in a parabolic diagonalizable stochastic equation driven by fractional noises. Based on the existence and uniqueness of the solution, we then obtain a kernel estimator of time-varying coeffcient and the convergence rates. An example is given to illustrate the theorem.     

Limsup results and LIL for partial sum processes of a Gaussian random field

Let {ξ _j ; j ∈ ℤ₊ ^d be a centered stationary Gaussian random field, where ℤ₊ ^d is the d-dimensional lattice of all points in d-dimensional Euclidean space ℝ^d, having nonnegative integer coordinates. For each j = (j ₁ , ..., jd) in ℤ₊ ^d , we denote |j| = j ₁ ... j _d and for m, n ∈ ℤ₊ ^d , define S(m, n] = Σ _m<j≤n ζ _j , σ²(|n−m|) = ES ² (m, n], S _n = S(0, n] and S ₀ = 0. Assume that σ(|n|) can be extended to a continuous function σ(t) of t > 0, which is nondecreasing and regularly varying with exponent α at b ≥ 0 for some 0 < α < 1. Under some additional conditions, we study limsup results for increments of partial sum processes and prove as well the law of the iterated logarithm for such partial sum processes. Research supported by NSERC Canada grants at Carleton University, Ottawa     

On the construction of nearly optimal strategies for a general problem of control of partially observed diffusions

In the paper a stochastic control problem consisting of continuously acting controls, impulse controls and stopping times of a partially observed diffusion with discounted cost functional is considered. The problem is successively approximated by time, observation, control and space discretizations to obtain finally a completely observed finite valued control problem. It is shown that the optimal strategy for the approximating problem, which can be numerically computed, is nearly optimal for the original problem     

Deterministic optimal control,given only a partial observation of the initial state

The deterministic linear-system, quadratic-cost optimal control problem is considered when the only state information available is a partial linear observation of the initial statex ₀. Thus, it is only known that the initial condition belongs to a particular linear variety. A control function is found which is optimal, in the sense (roughly) that (i) it can be computed using available information aboutx ₀ and (ii) no other control function which can be found using that information gives lower cost than it does for every initial condition that could have given rise to the information. The optimal control can be found easily from the conventional Riccati equation of optimal control. Applications are considered in the presence of unknown exponential disturbances and to the case with a sequence of partial state observations.     

On the limiting behavior of the maximum partial sums for arrays of rowwise NA random variables

Let {Xni, 1 ≤ n,i ＜∞} be an array of rowwise NA random variables and {an, n ≥ 1} a sequence of constants with 0 ＜ an ↑∞. The limiting behavior of maximum partial sums 1/an max 1≤k≤n| kΣi=1 Xni| is investigated and some new results are obtained. The results extend and improve the corresponding theorems of rowwise independent random variable arrays by Hu and Taylor [1] and Hu and Chang [2].     

Feasible command strategies for the control of motion of a simple mobile robot

The concept of feasible command strategies is introduced and its applicability is demonstrated by solving a guidance and control problem. This problem concerns the motion of a system which is composed of a rolling disk and a controlled slender rod that is pivoted, through its endpoint, about the disk center. The motion of the disk-rod system is subjected to state and control constraints, and it serves as a model for the motion of a simple mobile robot. In addition, the concept of path controllability is introduced and a condition is derived for the system motion path controllability. The derivation of this condition enables one to design closed-loop control laws for the system motion.     

OPTIMAL INTERIOR PARTIAL REGULARITY FOR NONLINEAR ELLIPTIC SYSTEMS UNDER THE NATURAL GROWTH CONDITION： THE METHOD OF A-HARMONIC APPROXIMATION

In this article,the authors consider the nonlinear elliptic systems under the natural growth condition.They use a new method introduced by Duzaar and Grotowski, for proving partial regularity for weak solutions,based on a generalization of the technique of harmonic approximation.And directly establish the optimal Holder exponent for the derivative of a weak solution.