Weak convergence of Markov-modulated diffusion processes with rapid switching

In this paper, we study the weak convergence of a sequence of Markov-modulated diffusion processes when the modulating Markov chain is ergodic and rapidly switching. We prove, in particular, its tightness property based on Aldous’ tightness criterion.     

Exponential convergence in ‐Wasserstein distance for diffusion processes without uniformly dissipative drift

By adopting the coupling by reflection and choosing an auxiliary function which is convex near infinity, we establish the exponential convergence of diffusion semigroups with respect to the standard ‐Wasserstein distance for all . In particular, we show that for the Itô stochastic differential equation if the drift term b is such that for any , holds with some positive constants K₁, K₂ and , then there is a constant such that for all , and , where is a positive constant. This improves the main result in 14 where the exponential convergence is only proved for the L¹‐Wasserstein distance.     

Weak convergence of diffusion processes on Wiener space

Let γ be a Gaussian measure on a Suslin space X, H be the corresponding Cameron–Martin space and {e _i } ⊂ H be an orthonormal basis of H. Suppose that μ _n = ρ _n · γ is a sequence of probability measures which converges weakly to a probability measure μ = ρ · γ Consider a sequence of Dirichlet forms , where and . We prove some sufficient conditions for Mosco convergence where . In particular, if X is a Hilbert space, and can be uniformly approximated by finite dimensional conditional expectations for every fixed e _i , then under broad assumptions Mosco and the distributions of the associated stochastic processes converge weakly.     

R~n上扩散过程的代数式收敛

本文研究n维欧氏空间上的扩散过程在L2意义下的代数式收敛的情况,给出了判定代数式收敛的方法,并对两种特殊情形扩散算子进行了讨论.将所得判敛法应用于两个例子可得到精确的结果.     

扩散过程代数式收敛比较判敛法

研究不同的马氏半群无穷小算子之间代数式收敛的关系,获得了若干比较定理.从而可将许多一般形式的扩散算子与特殊情形进行比较,将其化归为特殊的简单易算的情形.     

Stochastic differential inclusions and diffusion processes

Connections between weak solutions of stochastic differential inclusions and solutions of partial differential inclusions, generated by given set-valued mappings are considered. The main results are based on some continuous approximation selection theorem and weak compactness of the set of all weak solutions to a given stochastic differential inclusion.     

Algebraic convergence of function groups

We give a sufficient condition for a sequence of convex cocompact hyperbolic structures on a fixed compression body to have an algebraically convergent subsequence. This extends a result of Otal. Further if the manifold is a handlebody we show that certain laminations play a similar role in deformation space as binding curves in Teichmüller theory. Received: November 30, 2000     

The approximation of additive functionals of diffusion processes

Let us consider a diffusion process in R_d . Around each point x one may consider a ring of size ? and a process which counts the crossings over the ring. Integrating with respect to a measure μ(dx) and letting ?→ 0 one gets an additive functional. This is a natural generalization of the approximation theorem of the local time of one dimensional Brownian motion by means of “downcrossings”. For multidimensional Brownian motion the result was established by Bally. The present paper introduces a new method which allows us to handle general diffusions     

On the rate of convergence of spatial birth-and-death processes

Sufficient conditions for geometrical fast convergence of general spatial birth-and-death processes to equilibrium are established.     

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 .     

Small random perturbations of one-dimensional diffusion processes

In the present paper we consider the small random perturbations of one-dimensional diffusion processes. By virtue of stochastic analysis methods, we investigate the asymptotics of the mean exit times and probabilistical estimates of the exit times as the perturbations tend to zero. Partially supported by the Young Teachers Foundation of Beijing Institute of Technology     

Conservativeness of diffusion processes with drift

We show the conservativeness of the Girsanov transformed diffusion process by drift with or 4d/(d+2)$">, or if is of the Hardy class with sufficiently small coefficient of energy . Here 0$"> is the lower bound of the symmetric measurable matrix-valued function appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by when .

     

Asymptotic convergence of solutions for Laplace reaction–diffusion equations

We study the initial–boundary value problem for a Laplace reaction–diffusion equation. After constructing local solutions by using the theory of abstract degenerate evolution equations of parabolic type, we show asymptotic convergence of bounded global solutions if they exist under the assumption that the reaction function is analytic in neighborhoods of their $\omega$-limit sets. Reduction of degenerate evolution equation to multivalued evolution equation enables us to use the theory of the infinite-dimensional Łojasiewicz–Simon gradient inequality.     

Rate of convergence in the central limit theorem for empirical processes

Let _n and be an empirical process and a generalized Brownian bridge, respectively, indexed by a class of real measurable functions. From the central limit theorem for empirical processes it follows that for allr0. In this paper, assuming the class to be countably determined, under certain conditions we obtain an estimate for some constantC. Vapnik-ervonenkis class and the indicators of lower left orthants provide examples of classes considered here.     

Coupling methods and exponential ergodicity for two-factor affine processes

In this paper, by invoking the coupling approach, we establish exponential ergodicity under the L¹-Wasserstein distance for two-factor affine processes. The method employed herein is universal in a certain sense so that it is applicable to general two-factor affine processes, which allow that the first component solves a general Cox-Ingersoll-Ross (CIR) process, and that there are interactions in the second component, as well as that the Brownian noises are correlated; and even to some models beyond two-factor processes.     

On the asymptotic behavior of the parameter estimators for some diffusion processes: application to neuronal models

We consider a sample of i.i.d. times and we interpret each item as the first-passage time (FPT) of a diffusion process through a constant boundary. The problem is to estimate the parameters characterizing the underlying diffusion process through the experimentally observable FPT’s. Recently in Ditlevsen and Lánsky (Phys Rev E 71, 2005) and Ditlevsen and Lánsky (Phys Rev E 73, 2006) closed form estimators have been proposed for neurobiological applications. Here we study the asymptotic properties (consistency and asymptotic normality) of the class of moment type estimators for parameters of diffusion processes like those in Ditlevsen and Lánsky (Phys Rev E 71, 2005) and Ditlevsen and Lánsky (Phys Rev E 73, 2006). Furthermore, to make our results useful for application instances we establish upper bounds for the rate of convergence of the empirical distribution of each estimator to the normal density. Applications are also considered by means of simulated experiments in a neurobiological context.      

Computing conditional expectations of multidimensional diffusion processes

We study multidimensional diffusion processes and give an explicit representation for their conditional expectation. Starting from the solution formula for one dimensional stochastic differential equations found in Lanconelli and Proske [8], we compute the conditional expectation of a certain class of multidimensional diffusions without resorting to the Markov property of the process and therefore without requiring an explicit expression for the semi group associated to it.     

Foci and Foliations of Real Algebraic Curves

We discuss diverse results whose common thread is the notion of focus of an algebraic curve. In a unified setting, which combines elements of projective geometry, complex analysis and Riemann surface theory, we explain the roles of ordinary and singular foci in results on numerical ranges of matrices, quadrature domains, Schwarzian reflection, and other topics. We introduce the notion of canonical foliation of a real algebraic curve, which places foci into the context of continuous families of plane curves and provides a useful method of visualization of all relevant structures in a planar graphical image. Lecture held by Joel Langer in the Seminario Matematico e Fisico on October 6, 2006 Received: July 2007     

Strong solutions of SDES with singular drift and Sobolev diffusion coefficients

In this paper we prove the existence of a unique strong solution up to the explosion time for an SDE with a uniformly non-degenerate Sobolev diffusion coefficient (non-Lipschtiz) and locally integrable drift coefficient. Moreover, two non-explosion conditions are given.