Conservativeness and Extensions of Feller Semigroups

Let denote a Feller semigroup on , and itsextension to the bounded measurable functions. We show that . If the generator of the semigroup is a pseudo-differential operator we can restate this condition in terms of the symbol. As a by-product, we obtain necessary and sufficient conditions for the conservativeness of the semigroup which are again expressed through the symbol.     

一类N参数Gauss过程的异常震动点集合的Hausdorff维数

引进了一类N参数Gauss过程,它具有比N参数Wiener过程更为一般的性质．给出了此类N参数Gauss过程的异常震动点集的定义,并且定义了此异常震动点集的Hausdorff维数．研究了此类过程的异常震动点集Hausdorff维数,给出了它的一个确切的表达式,从而获得了与Zacharie (2001)的有关两参数Wiener过程的类似的结果．考虑的参数点集是一般的超长方体．而不是Zacharie (2001)考虑的超正方体．在此更为一般的情况下,首先建立了文中引进的过程的Fernique不等式．利用此不等式和Slepian引理,证明了过程的Lévy连续模定理．Zacharie(2001)关于Hausdorff维数公式的证明依赖于两参数Wiener过程的独立增量性,而这里引进的过程不具有这种性质,因此,必须采用新的证明途径．     

Conservativeness of Semigroups Generated by Pseudo Differntial Operators

Assume that the pseudo differential operator -q(x,D) generates a Fellerian or sub-Markovian semigroup. Under some natural additional conditions on the symbol -q(x,) we prove that the operator -q(x,D) is conservative if and only if q(x,0)0.     

自相似马氏过程的Hausdorff维数(英文)

本文讨论了取值于 Rd的随机过程 Xt=( X1 ( t) ,X2 ( t) ,… ,XN( t) ) ,并且在一定条件下获得了 Xt的像集和图集的 Hausdorff维数 .这里 Xi( t) ,t∈ R 是独立的取值于 Rdi 的αi-自相似马氏过程 ,并且 ∑Ni=1 di=d.     

由表示系统生成的分形的维数

在这篇文章里,由Ｒｎ中点的表示系统所生成的自仿射集中,给出了自仿射集满足Ｍｏｒａｎ开集条件的一个新的判别方法和不满足开集条件的自相似集的Ｈａｕｓｄｏｒｆｆ维数的上界和下界,并根据两个实例估计出其上下界是相等的．     

Characteristic Functions and Symbols in the Theory of Feller Processes

We derive a probabilistic expression for the symbol of the generator of a Feller process.     

On the existence of Feller semigroups with discontinuous coefficients II

This paper is devoted to the functional analytic approach to the problem of existence of Markov processes with Dirichlet boundary condition, oblique derivative boundary condition and first-order Wentzell boundary condition for second-order, uniformly elliptic differential operators with discontinuous coefficients. More precisely, we construct Feller semigroups associated with absorption, reflection, drift and sticking phenomena at the boundary. The approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderon- Zygmund theory of singular integral operators with non-smooth kernels.     

Julia集及其Hausdorff维数的连续性

对于d≥2,考虑多项式族Pc=Zd+c,c∈C.Kc={z∈C|{Pcn(z)}n≥0有界}为Pc的填充Julia集,Jc=(?)Kc为其Julia集.HD(Jc)为Jc的Hausdorff维数.设ω(0)为Pc0的临界点0的轨道的聚点集.我们假定Pc0在ω(0)上是扩张的,且O∈Jc0,|c0|>ε>0.如果一序列Cn→c0,则Jcn→Jc0,Kcn→Jc0,在Hausdorff拓扑下.如果存在一常数C1>0和一序列cn→c0,使得d(cn,Jc0)≥C1|cn-c0|1+1/d,则HD(Jcn)→HD(Jc0).这里d(cn,Jc0)为cn与Jc0间距离.     

On the Existence of Feller Semigroups with Discontinuous Coefficients

This paper is devoted to the functional analytic approach to the problem of the existence of Markov processes in probability theory. More precisely, we construct Feller semigroups with Dirichlet conditions for second-order, uniformly elliptic integro-differential operators with discontinuous coefficients. In other words, we prove that there exists a Feller semigroup corresponding to such a diffusion phenomenon that a Markovian particle moves both by jumps and continuously in the state space until it dies at the time when it reaches the boundary.     

稠密的s维分形集

对任意给定的0≤s≤1,本文构造Cantor型集E_s,使dim_H E_s=s,且E_s在[0,1]内稠密。     

On the Hausdorff Dimension of Some Random Cantor Sets

We consider Cantor sets C _X,p constructed by means of a sequence {X _k ^–p } _k=1 of random variables. We find sufficient conditions for the inequality to hold.     

Function Spaces as Path Spaces of Feller Processes

Let {X_t}_{t ≥ 0} be a Feller process with infinitesimal generator (A, D(A)). If the test functions are contained in D(A), —A |C^∞_c (ℝⁿ) is a pseudo–differential operator p(x, D) withsymbol p(x, ξ). We investigate local and global regularity properties of the sample paths t ↦ X_t in terms of (weighted) Besov B^s_pq (ℝ, ρ) and Triebel–Lizorkin F^s_pq (ℝ, ρ) spaces. The parameters for these spaces are determined by certain indices that describe the asymptotic behaviour of the symbol p(x, ξ). Our results improve previous papers on Lévy [5, 9] and Feller processes [22].     

分形插值函数图象的Hausdorff维数

本文给出了插值点为的分形插值函数图象的Hausdorff维数的下界估计     

Hausdorff Dimension for Range and Graph of Multi-Parameter Operator Stable L\'{e}vy Processes

The lower Hausdorff dimension results for the range and the graph of multi-parameter operator stable L\'{e}vy processes are established. The consequences are completely determined by the eigenvalues of its exponent matrix.     

Hausdorff Dimension for Range and Graph of Multi-Parameter Operator Stable L\'{e}vy Processes

??The lower Hausdorff dimension results for the range and the graph of multi-parameter operator stable L\'{e}vy processes are established. The consequences are completely determined by the eigenvalues of its exponent matrix.     

On the Continuity of Julia Sets and Hausdorff Dimension of N CP and Parabolic Maps

Denote by HD(J(f))the Hansdorff dimension of the Julia set J(f)of a rational function f.Our first result asserts that if f is an NCP map,and f_n→f horocyclically, preserving sub-critical relations,then f_n is an NCP map for all n(?)0 and J(f_n)→J(f)in the Hausdorff topology.We also prove that if f is a parabolic map and f_n is an NCP map for all n(?)0 such that f_n→f horocyclically,then J(f_n)→J(f)in the Hansdorff topology, and HD(J(f_n))→HD(J(f)).     

On the Connection between the Order of Riemann-Liouvile Fractional Calculus and Hausdorff Dimension of a Fractal Function

This paper investigates the fractal dimension of the fractional integrals of a fractal function.It has been proved that there exists some linear connection between the order of Riemann-Liouvile fractional integrals and the Hausdorff dimension of a fractal function.     

Dissipative Dynamics for a Class of Nonlinear Pseudo-Differential Equations

We study a class of nonlinear evolutionary equations generated by an elliptic pseudo-differential operator, and with nonlinearity of the form G(u _x ) where cη² ≤ G(η) ≤ Cη² for large |η|. For the evolution in spaces of periodic functions with zero mean we demonstrate existence of a universal absorbing set and compact attractor. Furthermore, we show that the attractor is of a finite Hausdorf dimension. The dissipation mechanism for the class of equations studied in the paper is akin to the nonlinear saturation in the Kuramoto-Sivashinsky equation. A similar generalization of the Kuramoto-Sivashinsky equation was studied by Nicolaenko et al. under the assumption of a purely quadratic nonlinearity and reflection invariance of both: the equation and solutions.      

R~d中齐次Moran集的Hausdorff维数

本文利用位势理论给出了Ｒｄ中齐次Ｍｏｒａｎ集的Ｈａｕｓｄｏｒｆｆ维数公式,从而回答了山中的问题．