Markov链的瞬时态与Martin边界的个数

在本文中,我们证明了含有限瞬时态的Markov链的瞬时态的个数不超过它的Martin积极流出边界的个数。     

Martin Boundary of a Fractal Domain

A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack principle. We show that a certain self-similar fractal has its complement as a uniformly John domain. In particular, the complement of the 3-dimensional Sierpiski gasket is a uniform domain and its Martin boundary is homeomorphic to the Sierpiski gasket itself.     

Martin boundary and exit space on the Sierpinski gasket

We define a new Markov chain on the symbolic space representing the Sierpinski gasket (SG),and show that the corresponding Martin boundary is homeomorphic to the SG while the minimal Martin boundary is the three vertices of the SG.In addition,the harmonic structure induced by the Markov chain coincides with the canonical one on the SG.This suggests another approach to consider the existence of Laplacians on those self-similar sets for which the problem is still not settled.     

Martin Boundary of a Killed Random Walk on a Half-Space

A complete representation of the Martin boundary of killed random walks on a half-space ℤ ^d−1×ℕ^* is obtained. In particular, it is proved that the corresponding Martin boundary is homemorphic to the half-sphere . The method is based on a combination of ratio limits theorems and large deviation techniques.     

On the Limits at the Martin Boundary for a Class of Functions

In this paper we prove a criterion for existence of pathwise limits at the Martin boundary for functions with gradient in L _loc ². (This implies that such functions have fine limits at almost all Martin boundary points.)     

Approximation in the Cone of Positive Harmonic Functions

In the course of studying quadrature domains Gustafsson, Sakai and Shapiro were led to the question of whether it is the case that the positive integrable harmonic functions on a bounded domain are dense among all positive harmonic functions (w.r.t. uniform convergence on compact subsets). In this article we will show how such approximation problems are related to representing measures on the Martin boundary, and then we use these results to give a counterexample to the question posed above. This research was supported by Science Foundation Ireland under Grant 06/RFP/MAT057.     

The Dirichlet problem at the Martin boundary of a fine domain

We adapt the Perron–Wiener–Brelot method of solving the Dirichlet problem at the Martin boundary of a Euclidean domain so as to cover also the Dirichlet problem at the Martin boundary of a fine domain U in ${\mathbb{R}}^n$ ($n\ge 2$) (i.e., a set U which is open and connected in the H. Cartan fine topology on ${\mathbb{R}}^n$, the coarsest topology in which all superharmonic functions are continuous). It is a complication that there is no Harnack convergence theorem for so-called finely harmonic functions. We define resolutivity of a numerical function on the Martin boundary $\mathrm{\Delta }(U)$ of U. Our main result Theorem 4.14 implies the corresponding known result for the classical case. We also obtain analogous results for the case where the upper and lower PWB-classes are defined in terms of the minimal-fine topology on the Riesz–Martin space $\stackrel{￣}{U}=U\cup \mathrm{\Delta }(U)$ instead of the natural topology. The two corresponding concepts of resolutivity are compatible.     

Boundary Behaviour of Positive Solutions of Second-Order Parabolic Operators with Lower-Order Terms in a Half-Space

In this paper, we study the boundary behaviour of positive solutions of certain parabolic operators with lower order terms in a half-space. Basing on these results, we characterize the Martin boundary and show that any positive solution has an integral representation. We thus generalize to a large class of parabolic operators some results proved by Kaufman, Wu, Mair, and Taylor for the heat equation.     

Martin Boundaries of Quasi-Sectorial Domains

Cranston and Salisbury have obtained an integral test for the existence of a maximal number of minimal Martin boundary points at 0 in certain domains (cf. [8]). This paper will extend the result as follows: let D R ^d be an open Greenian set (with respect to the Laplacian) consisting of n disjoint open connected cones with Lipschitz boundary and a subset of the boundary of these cones. Let be some local Kato measure supported by the boundary of the cones and consider the Schrödinger operator 1/2-µ. We will assume a boundary Harnack principle and give a sufficient integral criterion for the existence of exactly n minimal Martin boundary points at 0. In certain cases there is a necessary criterion, too. When the sufficient integral criterion holds we will give a necessary and a sufficient condition for the existence of a certain process related to the Schrödinger operator that connects two different admissible boundary points. In the paper of Cranston and Salisbury the case = 0, d = 2 is treated, but many of the arguments work as well in the general situation.     

Combinatorics of rooted trees and Hopf algebras

We begin by considering the graded vector space with a basis consisting of rooted trees, with grading given by the count of non-root vertices. We define two linear operators on this vector space, the growth and pruning operators, which respectively raise and lower grading; their commutator is the operator that multiplies a rooted tree by its number of vertices, and each operator naturally associates a multiplicity to each pair of rooted trees. By using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint, and obtain several results about the associated multiplicities.

Now the symmetric algebra on the vector space of rooted trees (after a degree shift) can be endowed with a coproduct to make a Hopf algebra; this was defined by Kreimer in connection with renormalization. We extend the growth and pruning operators, as well as the inner product mentioned above, to Kreimer's Hopf algebra. On the other hand, the vector space of rooted trees itself can be given a noncommutative multiplication: with an appropriate coproduct, this leads to the Hopf algebra of Grossman and Larson. We show that the inner product on rooted trees leads to an isomorphism of the Grossman-Larson Hopf algebra with the graded dual of Kreimer's Hopf algebra, correcting an earlier result of Panaite.

     

粘滑混合边界条件下平面边界前的Stokes流动

对具有粘滑混合边界条件的平面边界,建立一个Stokes流动的一般性定理,利用双调和函数A与调和函数B,表示了3维Stokes流动的速度场和压力场.关于无滑动平面边界前Stokes流动的早期定理,成为该一般性定理的一个特例.进一步地,从一般性定理导出了一个推论,根据该Stokes流函数,给出了粘滑边界条件时刚性平面轴对称Stokes流动问题的解,得到了流体作用在边界上的牵引力和扭矩公式.给出了一个说明性的例子.     

关于换位子范数的若干不等式

当f是[0,∞)上的非负算子的单调函数时,得到了对任何非交换Banach函数空间范数都有‖f (A)X - Xf (B)‖ 54‖f ( AX - XB )‖,其中A,B是τ-可测正算子, X是收缩算子.     

Integration of vector Sobolev-type PDE over octonions

Vector mixed Sobolev-type partial differential equations (PDE) are studied. The technique of their integration over the octonion algebra is developed. Theorems about integration of vector mixed Sobolev-type PDE over octonions are proved.     

Asymptotics for Certain Harmonic Functions and the Martin Compactification on the Quaternionic Heisenberg Group

In this paper, we make the asymptotic estimates of the heat kernel for the quaternionic Heisenberg group in various cases. We also use these results to deduce the asymptotic estimates of certain harmonic functions on the quaternionic Heisenberg group. Moreover a Martin compactification of the quaternionic Heisenberg group is constructed, and we prove that the Martin boundary of this group is homeomorphic to the unit ball in the quaternionic field.     

最小Q-过程中断时Martin流出边界与Ray-Knight紧化的关系

在最小Q-过程中断条件下证明了Martin流出边界与Ray-Knight紧化之间存在一定的映射关系.并举例说明如果最小Q-过程中断且无限流出,则Ray-Knight紧化可能比Martin流出边界要多出一些点.因此在Markov链的构造论中不能只考虑Martin边界.     

Potential Theory of Special Subordinators and Subordinate Killed Stable Processes

In this paper we introduce a large class of subordinators called special subordinators and study their potential theory. Then we study the potential theory of processes obtained by subordinating a killed symmetric stable process in a bounded open set D with special subordinators. We establish a one-to-one correspondence between the nonnegative harmonic functions of the killed symmetric stable process and the nonnegative harmonic functions of the subordinate killed symmetric stable process. We show that nonnegative harmonic functions of the subordinate killed symmetric stable process are continuous and satisfy a Harnack inequality. We then show that, when D is a bounded κ-fat set, both the Martin boundary and the minimal Martin boundary of the subordinate killed symmetric stable process in D coincide with the Euclidean boundary ∂D. The research of this author is supported in part by MZOS grant 0037107 of the Republic of Croatia and in part by a joint US-Croatia grant INT 0302167.     

边界过原点的任意半平面中的Hilbert边值问题

给出了边界过原点的任意半平面中的Hilbert边值问题的提法,定义了函数的一种对称扩张,并利用这种对称扩张将此Hilbert边值问题转化为无穷直线上的Riemann边值问题,得到了该问题的一般解和可解性定理.