On the diffusion equation and diffusion wavelets on flat cylinders and the n‐torus

In this paper we study the solutions to the diffusion equation on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the parabolic Dirac operator. We study their fundamental properties, give representation formulas of all these solutions and develop some integral representation formulas. In particular we set up a Green type formula for the solutions to the homogeneous diffusion equation on cylinders and tori. Then we also treat the inhomogeneous diffusion equation diffusion with prescribed boundary conditions in Lipschitz domains on these manifolds. As main application, we construct well localized diffusion wavelets on this class of cylinders and tori by means of multiperiodic eigensolutions to the parabolic Dirac operator. We round off with presenting some concrete numerical simulations for the three dimensional case. Copyright © 2010 John Wiley & Sons, Ltd.     

The strong weak convergence of the quasi-EA

In this paper, we investigate the convergence of a novel simulation scheme to the target diffusion process. This scheme, the Quasi-EA, is closely related to the Exact Algorithm (EA) for diffusion processes, as it is obtained by neglecting the rejection step in EA. We prove the existence of a myopic coupling between the Quasi-EA and the diffusion. Moreover, an upper bound for the coupling probability is given. Consequently we establish the convergence of the Quasi-EA to the diffusion with respect to the total variation distance.     

A new inverse problem for the diffusion operator

In this work, we have estimated nodal points and nodal lengths for the diffusion operator. Furthermore, by using these new spectral parameters, we have shown that the potential function of the diffusion operator can be established uniquely. An analogous inverse problem was solved for the Sturm–Liouville problem in recent years.     

分数扩散测度的分部积分公式及鞅表示定理

本文研究由分数扩散过程决定的测度(分数扩散测度)的随机分析理论.首先,利用Bismut方法给出拉回公式,得到了分数扩散测度的分部积分公式.进一步,利用此公式,将Wiener测度下的经典的鞅表示定理推广到分数扩散测度下的鞅表示定理.     

扩散模型中扩散系数的非参数组合估计

为了提高扩散系数估计的准确度, 我们利用动态组合时间域与状态域信息提出一个新的组合估计量. 我们发现所提组合估计量能有效估计扩散模型的扩散系数, 正如在本文中模拟所示. 在一定的条件下, 建立了估计量的渐进正态性, 并证明了时间域估计量与状态域估计量是渐进独立的. 大量的模拟展示了所提组合估计量优于单域估计量, 也优于本文所提估计量.     

On the strong convergence rate for the Euler–Maruyama scheme of one-dimensional SDEs with irregular diffusion coefficient and local time

The strong rate of convergence for the Euler–Maruyama scheme of stochastic differential equations (SDEs) driven by a Brownian motion with Hölder continuous diffusion coefficient or irregular drift coefficient have been widely studied. In the case of irregular diffusion coefficient, however, there are few studies. In this article, under Le Gall's condition on the diffusion coefficient, which leads to conclude the pathwise uniqueness for SDEs, we provide the same result on the strong rate of convergence as in the case of 1/2-Hölder continuous diffusion coefficient. The idea of the proof is to use a version of Avikainen's inequality. As an application, we introduce a numerical scheme for SDEs with local time.     

一个非线性扩散系统解的存在性及线性系统的最优控制

讨论关于生物种群的一个非线性扩散系统和线性系统的一些问题,得到了非线性扩散系统弱解的存在性;线性系统最优繁殖率的存在性和关于边界扰动的最优解的存在唯一性。     

Tikhonov regularization method for a backward problem for the inhomogeneous time-fractional diffusion equation

Fractional (nonlocal) diffusion equations replace the integer-order derivatives in space and time by their fractional-order analogs and they are used to model anomalous diffusion, especially in physics. In this paper, we study a backward problem for an inhomogeneous time-fractional diffusion equation with variable coefficients in a general bounded domain. Such a backward problem is of practically great importance because we often do not know the initial density of substance, but we can observe the density at a positive moment. The backward problem is ill-posed and we propose a regularizing scheme by using Tikhonov regularization method. We also prove the convergence rate for the regularized solution by using an a priori regularization parameter choice rule. Numerical examples illustrate applicability and high accuracy of the proposed method.     

On the well‐posedness of a mathematical model of quorum‐sensing in patchy biofilm communities

We analyze a system of reaction–diffusion equations that models quorum‐sensing in a growing biofilm. The model comprises two nonlinear diffusion effects: a porous medium‐type degeneracy and super diffusion. We prove the well‐posedness of the model. In particular, we present for the first time a uniqueness result for this type of problem. Moreover, we illustrate the behavior of model solutions in numerical simulations. Copyright © 2011 John Wiley & Sons, Ltd.     

Wavelet estimation of the diffusion coefficient in time dependent diffusion models

The estimation problem for diffusion coefficients in diffusion processes has been studied in many papers,where the diffusion coefficient function is assumed to be a 1-dimensional bounded Lipschitzian function of the state or the time only.There is no previous work for the nonparametric estimation of time-dependent diffusion models where the diffusion coefficient depends on both the state and the time.This paper introduces and studies a wavelet estimation of the time-dependent diffusion coefficient under a more general assumption that the diffusion coefficient is a linear growth Lipschitz function.Using the properties of martingale,we translate the problems in diffusion into the nonparametric regression setting and give the L~r convergence rate.A strong consistency of the estimate is established.With this result one can estimate the time-dependent diffusion coefficient using the same structure of the wavelet estimators under any equivalent probability measure.For example, in finance,the wavelet estimator is strongly consistent under the market probability measure as well as the risk neutral probability measure.     

空气中PM2.5问题的研究

主要研究空气污染中的PM2.5扩散问题.首先利用相关分析法讨论了PM2.5与SO_2,NO_2,CO,PM10,O_3的相关性,建立线性回归方程;然后建立一维的反应扩散方程,预测PM2.5浓度变化,并定量与定性分析西安市空气污染状况;再建立高斯烟羽模型,对持续高浓度PM2.5扩散情形进行拟合,并对污染物扩散范围进行预测,得到重度污染以及可能安全区域;最后通过建立最优化模型,得到较有经济效益的空气治理方案.     

A convergence result for the periodic unfolding method related to fast diffusion on manifolds

Based on the periodic unfolding method in periodic homogenization, we deduce a convergence result for gradients of functions defined on connected, smooth, and periodic manifolds. Under the assumption of certain a-priori estimates of the gradient, which are typical for fast diffusion, the sum of a term involving a gradient with respect to the slow variable and one with respect to the fast variable is obtained in the homogenization limit. In addition, we show in a brief example how to apply this result and find for a reaction–diffusion equation defined on a periodic manifold that the homogenized equation contains a term describing macroscopic diffusion.     

The coherent potential method in the diffusion problem on a random substitution lattice

Based on the balance equation, we consider the diffusion problem on a hyperlattice with randomly distributed inaccessible sites. Using diagram methods, we find a self-consistent expression for the configurationally averaged Green’s function in the coherent potential approximation. We show that this approach is applicable in a broad range of concentrations of accessible sites. Using this approximation, we find the exact asymptotic form of the static diffusion coefficient for a low concentration of blocked sites. This allows making good estimates of the percolation threshold in the random-site diffusion problem on an arbitrary hyperlattice. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 149, No. 2, pp. 252–261, November, 2006.     

Smoothness of the Distribution of the Supremum of a Multi-dimensional Diffusion Process

In this article we deal with a multi-dimensional diffusion whose corresponding diffusion vector fields are commutative, and study its joint distribution at the time when a component attains its maximum on finite time interval. Under regularity and ellipticity conditions we shall show the smoothness of this joint distribution.     

An operator-based approach to the analysis of ruin-related quantities in jump diffusion risk models

Recent developments in ruin theory have seen the growing popularity of jump diffusion processes in modeling an insurer’s assets and liabilities. Despite the variations of technique, the analysis of ruin-related quantities mostly relies on solutions to certain differential equations. In this paper, we propose in the context of Lévy-type jump diffusion risk models a solution method to a general class of ruin-related quantities. Then we present a novel operator-based approach to solving a particular type of integro-differential equations. Explicit expressions for resolvent densities for jump diffusion processes killed on exit below zero are obtained as by-products of this work.     

一类重入型网络在优先服务原则下的扩散近似

本文研究了一类重入型网络在优先服务原则下的扩散近似,运用随机分析方法,证明了标准化队长过程的C-紧性．在优先服务原则下,给出了这类网络的标准化队长过程扩散近似存在的充分条件．     

Analysis on the critical speed of traveling waves

The note is concerned with a time-delayed reaction–diffusion equation with nonlocality for the population dynamics of single species. For the critical speed of traveling waves, we give a detailed analysis on its location and asymptotic behavior with respect to the parameters of the diffusion rate and mature age, respectively.     

具有振动边值的扩散方程Dirichlet问题

本文以上（下）连续函数作为扩散方程u_t=1/2Δu+cu 在Ｄ内的Ｄｉｒｉｃｈｌｅｔ问题边值函数，讨论了振动边值的Ｄｉｒｉｃｈｌｅｔ问题，并用概率方法证明解的存在性、唯一性和稳定性，把古典Ｄｉｒｉｃｈｌｅｔ问题边值条件减弱到最一般情形     

The influence of a metasolution on the behaviour of the logistic equation with nonlocal diffusion coefficient

In this paper we use the bifurcation method and fixed point arguments to study a logistic equation with nonlocal diffusion coefficient. We prove the existence of an unbounded continuum of positive solutions that bifurcates from the trivial solution. The global behaviour of this continuum depends strongly on the value of the nonlocal diffusion coefficient at infinity as well as the relative position between the refuge of the species and the weight of the diffusion coefficient. Moreover, we show the complexity of the structure of the set of positive solutions using fixed point arguments.