Nonlinear Diffusion Processes

We study elliptic systems of strongly nonlinear first-order differential equations in complex form on the plane. For such systems we develop the theory of Hilbert boundary value problems which is very much similar to the well-known theory for a holomorphic vector. Systems of nonlinear elliptic equations describe problems of interaction of several nonlinear stationary processes in the diffusive and convective mass and heat transport by hydrodynamic fluid flows.     

RESEARCH ANNOUNCEMENTS——Coupled Diffusion Processes

Coupled diffusion processes (or CDP for short) model the systems of molecular motors, which attract much interest from physicists and biologists in recent years. The pro-tein moves along a filament called the track, and it is crucial that there are several inner states of the protein and the u‘nderlying chemical reaction causes transitions among different inner states, while chemical energy can be converted to mechanical energy by rachet effects. In this sense, the protein becomes a motor and is called a Brownian particle coupled with a chemical reaction. The impact of the chemical reaction can be considered as a particle source, hencethe Fokker-Planck equation should be     

反射扩散过程的极集

本文利用Ｄｉｒｉｃｈｌｅｔ型中有产在容度的结论，从几何的观点刻画了反射扩散过程的极集。     

Strong Stationary Duality for Diffusion Processes

We develop the theory of strong stationary duality for diffusion processes on compact intervals. We analytically derive the generator and boundary behavior of the dual process and recover a central tenet of the classical Markov chain theory in the diffusion setting by linking the separation distance in the primal diffusion to the absorption time in the dual diffusion. We also exhibit our strong stationary dual as the natural limiting process of the strong stationary dual sequence of a well-chosen sequence of approximating birth-and-death Markov chains, allowing for simultaneous numerical simulations of our primal and dual diffusion processes. Lastly, we show how our new definition of diffusion duality allows the spectral theory of cutoff phenomena to extend naturally from birth-and-death Markov chains to the present diffusion context.     

Angular Structure of Diffraction Processes

We discuss the general rules for the behavior of angular distributions in inelastic diffraction dissociation and differences in the angular structure of elastic and inelastic diffraction processes.     

非局部时滞反应扩散方程行波解的稳定性

考虑了非局部时滞反应扩散方程行波解的稳定性.在合适的加权L~∞空间下,证明了非临界波指数稳定,临界波代数稳定(即代数衰减).还利用加权能量估计的方法得到了衰减速率.我们把相关结果应用到了host-vector模型,Nicholson blowflies方程和一个修改了的传染病模型.     

The Approach of Solutions for the Nonlocal Diffusion Equation to Traveling Fronts

The paper is concerned with the asymptotic behavior as t → ±∞ of an entire solution u(x,t) for the nonlocal diffusion equation. With bistable assumption, it is well known that the model has three different types of travel- ing fronts. Under certain conditions on the wave speeds, and by some auxiliary rational functions with certain properties to construct appropriate super- and sub solutions of the model, we establish two new types of entire solutions u(x,t) which approach to three travelling fronts or the positive equilibrium as t → ±∞.     

Efficient Density Estimation for Ergodic Diffusion Processes

The problem of asymptotically efficient estimation of the density of invariant measure of a diffusion process is considered. The efficient estimator is defined with the help of the minimax lower bound on the risk of all estimators. We show that the local–time and kernel–type estimators are asymptotically efficient for the loss functions with polynomial majorants. The asymptotic behavior of a wide class of unbiased estimators with the same limit variances is also discussed. This revised version was published online in June 2006 with corrections to the Cover Date.     

流形上扩散过程的基本耦合

本文给出了光滑流形上扩散过程耦合算子的一般形式,并证明了一类耦合具有耦合的基本性质。     

Order－preservation and Positive Correlations for Diffusion Processes on Torus

Ｏｒｄｅｒ－ｐｒｅｓｅｒｖａｔｉｏｎａｎｄＰｏｓｉｔｉｖｅＣｏｒｒｅｌａｔｉｏｎｓｆｏｒＤｉｆｆｕｓｉｏｎＰｒｏｃｅｓｓｅｓｏｎＴｏｒｕｓ￥ＷａｎｇＦｅｎｇｙｕ（王凤雨）ａｎｄＹａｎＳｈｉｊｉｎｎ（严士健）（ＤｅｐａｒｔｍｅｎｔｏｆＭａｔｈｅｍａｔｉ...     

扩散过程代数式收敛定性的判别准则

本文定性地讨论非紧空间中可逆扩散过程的代数式收敛的判定 .使用分裂空间的方法 .将全空间分裂成两个部分 :紧的子空间与非紧的余子空间 .在紧子空间中考虑边界反射的Neumann过程 ,它必然是代数式收敛的 .而在非紧子空间中考虑边界吸收的Dirichlet过程 ,如果这一Dirichlet过程以代数式的速度击中边界 ,那么就有原过程在全空间代数式收敛 ;反之 ,原过程代数式收敛 ,非紧子空间中的Dirichlet过程也是代数式收敛的 .因此过程在紧子空间的任意摄动不会影响在全空间的代数式收敛性 .     

关于多维扩散过程耦合的保序性

本文给出了多维扩散过程耦合保序的若干充分条件和必要条件，特别地，当耦合过程存在唯一时，得到了耦合保序的充要条件。     

Asymptotics of Intersection Local Time for Diffusion Processes

In the paper, we investigate the intersection local time for two correlated Brownian motions on the plane that form a diffusion process in ?⁴ associated with a divergence-form generator. Using Gaussian heat kernel bounds, we prove the existence of intersection local time for these Brownian motions, obtain estimates of its moments, and establish the law of iterated logarithm for it.

     

Stochastic Taylor Expansions for Functionals of Diffusion Processes

In this article, a stochastic Taylor expansion of some functional applied to the solution process of an Itô or Stratonovich stochastic differential equation with a multi-dimensional driving Wiener process is given. Therefore, the multi-colored rooted tree analysis is applied in order to obtain a transparent representation of the expansion which is similar to the B-series expansion for solutions of ordinary differential equations in the deterministic setting. Further, some estimates for the mean-square and the mean truncation errors are given.     

On Multidimensional Generalized Diffusion Processes

We construct a multidimensional generalized diffusion process with the drift coefficient that is the (generalized) derivative of a vector-valued measure satisfying an analog of the Hölder condition with respect to volume. We prove the existence and continuity of the density of transition probability of this process and obtain standard estimates for this density. We also prove that the trajectories of the process are solutions of a stochastic differential equation.     

First Passage Densities and Boundary Crossing Probabilities for Diffusion Processes

We consider the boundary crossing problem for time-homogeneous diffusions and general curvilinear boundaries. Bounds are derived for the approximation error of the one-sided (upper) boundary crossing probability when replacing the original boundary by a different one. In doing so we establish the existence of the first-passage time density and provide an upper bound for this function. In the case of processes with diffusion interval equal to ℝ this is extended to a lower bound, as well as bounds for the first crossing time of a lower boundary. An extension to some time-inhomogeneous diffusions is given. These results are illustrated by numerical examples.