随机分形

本文概括了随机分形的主要结果，综述了随机分形的最新进展和目前的动态，提出了一些末解决的问题，全文共分为三部分：（１）由随机过程和随机场（如Ｌｅｖｙ过程，Ｇａｕｓｓ场，自相似过程等）产生的各种随机分形集（如象集、水平集、Ｋ重点集等）的Ｈａｕｓｄｏｒｆｆ维数、测度和ｐａｃｋｉｎｇ维数、测试；（２）随机Ｃａｎｔｏｒ型集和统计自相似集的维数和测试；（３）分形集（如Ｓｐｉｅｒｐｉｎｓｋｉ ｇａｓｋｅｔ，ｎｅ     

相关带干扰风险模型的破产概率

本文引进了含相关类带干扰经典风险过程, 研究类之间的相关性对破产概率的影响, 主要研究类之间的相关性对其Lundberg指数的大小关系的影响.     

Random Attractor for the Nonclassical Diffusion Equation with Fading Memory

In this paper,we consider the stochastic nonclassical diffusion equationwith fading memory on a bounded domain. By decomposition of the solution operator, we give the necessary condition of asymptotic smoothness of the solution to the initial boundary value problem, and then we prove the existence of a random attractor in the space $M_1=D(A^{\frac{1}{2}}) × L^2_μ(R^+, D(A^{\frac{1}{2}}))$, where A=-Δ with Dirichlet boundary condition.     

Dirichlet Forms and Brownian Motion Penetrating Fractals

Can a Brownian motion penetrate the two-dimensional Sierpinski gasket? This question was studied in [8], and an affirmative answer was given. In this paper, the problem is studied with a different approach, using Dirichlet forms and function space theory. The results obtained are somewhat different from, and from certain aspects more general than, the results in [8].

     

Packing Measure and Dimension of Random Fractals

We consider random fractals generated by random recursive constructions. We prove that the box-counting and packing dimensions of these random fractals, K, equals , their almost sure Hausdorff dimension. We show that some almost deterministic conditions known to ensure that the Hausdorff measure satisfies also imply that the packing measure satisfies 0< . When these conditions are not satisfied, it is known . Correspondingly, we show that in this case , provided a random strong open set condition is satisfied. We also find gauge functions (t) so that the -packing measure is finite.     

关于分数布朗运动预测的研究

本文对赫斯特参数H∈(1/2,1)的分数布朗运动的预测过程的样本轨道性质进行了讨论.利用布朗运动的随机积分理论,建立了一个重要的不等式,证明了(Z)的图集的Hausdorff维数等于1,得出了预测过程与分数布朗运动本身有显著不同特征的结论.     

A probabilistic approach to heat diffusion on symmetric spaces

We give an interpretation of heat diffusion phenomena on symmetric spaces as described by Anker and Setti⁽²⁾ in terms of the asymptotic behavior of Brownian motion.     

Metrics of Hyperbolic Type on Bounded Fractals

On the bounded Sierpinski gasket F we use the set of essential fixed points V ₀ as a boundary and consider the fractal Brownian motion on F killed in V ₀. The corresponding Dirichlet–Laplacian is described in terms of a kind of hyperbolic distance, a metric which explodes near the boundary. We consider Harnack inequalities, Green’s function estimates and (random) products of matrices defining the local energy of harmonic functions. Supported by the DFG research group ‘Spektrale Analysis, asymptotische Verteilungen und stochastische Dynamik.’     

Estimation of Drift Parameter and Change Point for Switching Fractional Diffusion Processes

We study the asymptotic distribution of the maximum likelihood estimator for the drift parameter and the change point for fractional diffusion processes as the noise intensity tends to zero.     

Exponential Convergence in Probability for Empirical Means of Brownian Motion and of Random Walks

Given a Brownian motion (B _t) _t0 in R ^d and a measurable real function f on R ^d belonging to the Kato class, we show that 1/t ₀ ^t f(B _s ) ds converges to a constant z with an exponential rate in probability if and only if f has a uniform mean z. A similar result is also established in the case of random walks.     

Time-Localization of Random Distributions on Wiener Space II: Convergence,Fractional Brownian Density Processes

For a random element X of a nuclear space of distributions on Wiener space C([0,1],R ^d ), the localization problem consists in projecting X at each time t[0,1] in order to define an S(R ^d )-valued process X={X(t),t[0,1]}, called the time-localization of X. The convergence problem consists in deriving weak convergence of time-localization processes (in C([0,1],S(R ^d )) in this paper) from weak convergence of the corresponding random distributions on C([0,1],R ^d ). Partial steps towards the solution of this problem were carried out in previous papers, the tightness having remained unsolved. In this paper we complete the solution of the convergence problem via an extension of the time-localization procedure. As an example, a fluctuation limit of a system of fractional Brownian motions yields a new class of S(R ^d )-valued Gaussian processes, the fractional Brownian density processes.     

紧流形上非退化扩散过程的成功耦合

本文构造了紧连通流形上非退化扩散过程的一些耦合，并使用随机分析的方法证明这些耦合是成功的，进一步地，耦合时间一阶矩有限。最后，我们使用耦合方法研究了扩散过程依全变差范数的收敛速度。     

Diffusion processes on complete riemannian manifolds

ＤＩＦＦＵＳＩＯＮＰＲＯＣＥＳＳＥＳＯＮＣＯＭＰＬＥＴＥＲＩＥＭＡＮＮＩＡＮＭＡＮＩＦＯＬＤＳＱＩＡＮＺＨＯＮＧＭＩＮ（钱忠民）（ＤｅｐａｒｔｍｅｎｔｏｆＡｐｐｌｉｅｄＭａｔｈｅｍａｔｉｃｓ,ＳｈａｎｇｈａｉＩｎｓｔｉｔｕｔｅｏｆＲａｉｌｗａｙＴｅｃ...     

Killed Random Processes and Heat Kernels

Let V(x) ≥ 0 be a given function tending to a constant at infinity. It is well known that the density of the Brownian motion B_t killed at the infinitesimal rate V is a Green's function for the heat operator with such a potential. With an appropriate generalization, its Laplace transform also gives the density of ∫ ₀ ^t V(B_s)ds. We construct such a Green's function via spectral analysis of the classical one-dimensional stationary Schrodinger operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 423–432, August, 2005. An erratum to this article is available at .     

A martingale characterization of the set-indexed Brownian motion

We characterize a Brownian motion indexed by a semilattice of sets, using the theory of set-indexed martingales: a square integrable continuous set-indexed strong martingale is a Brownian motion if and only if its compensator is deterministic and continuous.Research supported by a grant from the Natural Sciences and Engineering Research Council of Canada.Research done while this author was visiting the University of Ottawa. He wishes to thank Professor Ivanoff for her kind hospitality.     

Diffusion approximation of a multitype re-entrant line under smaller-buffer-first-served policy

This paper studies a multitype re-entrant line under smaller-buffer-first-served policy, which is an extension of first-buffer-first-served re-entrant line. We prove a heavy traffic limit theorem. The key to the proof is to prove the uniform convergence of the corresponding critical fluid model.     

On the Linear Fractional Self-attracting Diffusion

In this paper, we introduce the linear fractional self-attracting diffusion driven by a fractional Brownian motion with Hurst index 1/2<H<1, which is analogous to the linear self-attracting diffusion. For 1-dimensional process we study its convergence and the corresponding weighted local time. For 2-dimensional process, as a related problem, we show that the renormalized self-intersection local time exists in L ² if 1/2<H<3/4. The Project-sponsored by NSFC (10571025) and the Key Project of Chinese Ministry of Education (No.106076).     

Killed Random Processes and Heat Kernels [Erratum]

In J. Villarroel “Killed random processes and heat kernels” (Vol. 144, No. 2, August, 2005) on page 1238, the sentence before (2) should be: “We suppose that given that ( took a value x and hence has not yet been killed), the probability of being killed at any time t + h > t is O(h) ....” On page 1239, the expression for ζ(t; p|x′) after Eq. (6) should be:

Diffusion approximation forGI/G/1 controlled queues

A queueing model is considered in which a controller can increase the service rate. There is a holding cost represented by functionh and the service cost proportional to the increased rate with coefficientl. The objective is to minimize the total expected discounted cost.Whenh andl are small and the system operates in heavy traffic, the control problem can be approximated by a singular stochastic control problem for the Brownian motion, namely, the so-called reflected follower problem. The optimal policy in this problem is characterized by a single numberz ^* so that the optimal process is a reflected diffusion in [0,z ^*]. To obtainz ^* one needs to solve a free boundary problem for the second order ordinary differential equation. For the original problem the policy which increases to the maximum the service rate when the normalized queue-length exceedsz ^* is approximately optimal.