Boundary Harnack Principle and Martin Boundary at Infinity for Subordinate Brownian Motions

In this paper we study the Martin boundary of unbounded open sets at infinity for a large class of subordinate Brownian motions. We first prove that, for such subordinate Brownian motions, the uniform boundary Harnack principle at infinity holds for arbitrary unbounded open sets. Then we introduce the notion of κ-fatness at infinity for open sets and show that the Martin boundary at infinity of any such open set consists of exactly one point and that point is a minimal Martin boundary point.     

The Boundary Harnack Principle for Non-Divergence form Elliptic Operators

If L is a uniformly elliptic operator in non-divergence form,the boundary Harnack principle for the ratio of positive L-harmonicfunctions holds in Hölder domains of order if > . A counterexample shows that is sharp. For Hölder domainsof order with (0,1], the boundary Harnack principle holds providedthe domain also satisfies a strong uniform regularity condition.     

Transition Densities and Traces for Invariant Feller Processes on Compact Symmetric Spaces

We find necessary and sufficient conditions for a finite K–bi–invariant measure on a compact Gelfand pair (G,K) to have a square–integrable density. For convolution semigroups, this is equivalent to having a continuous density in positive time. When (G,K) is a compact Riemannian symmetric pair, we study the induced transition density for G–invariant Feller processes on the symmetric space X = G/K. These are obtained as projections of K–bi–invariant Lévy processes on G, whose laws form a convolution semigroup. We obtain a Fourier series expansion for the density, in terms of spherical functions, where the spectrum is described by Gangolli’s Lévy–Khintchine formula. The density of returns to any given point on X is given by the trace of the transition semigroup, and for subordinated Brownian motion, we can calculate the short time asymptotics of this quantity using recent work of Bañuelos and Baudoin. In the case of the sphere, there is an interesting connection with the Funk–Hecke theorem.     

The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form

Potential Analysis - We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that...     

Harnack inequality and strong Feller property for stochastic fast-diffusion equations

As a continuation to [F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab. 35 (2007) 1333-1350], where the Harnack inequality and the strong Feller property are studied for a class of stochastic generalized porous media equations, this paper presents analogous results for stochastic fast-diffusion equations. Since the fast-diffusion equation possesses weaker dissipativity than the porous medium one does, some technical difficulties appear in the study. As a compensation to the weaker dissipativity condition, a Sobolev-Nash inequality is assumed for the underlying self-adjoint operator in applications. Some concrete examples are constructed to illustrate the main results.     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

The Euler Scheme for Feller Processes

We consider the Euler scheme for stochastic differential equations with jumps, whose intensity might be infinite and the jump structure may depend on the position. This general type of SDE is explicitly given for Feller processes and a general convergence condition is presented.

In particular, the characteristic functions of the increments of the Euler scheme are calculated in terms of the symbol of the Feller process in a closed form. These increments are increments of Lévy processes and, thus, the Euler scheme can be used for simulation by applying standard techniques from Lévy processes.     

Approximation of Boundary Conditions at Infinity for a Harmonic Equation

Starting from the canonical boundary reduction, this paper studies an approximate differential boundary condition and an approximate integral boundary condition on an artificial boundary for the exterior problem of a harmonic equation, and gives an error estimate for the latter. This estimate reveals the relationship between the error and the approximate grade boundary conditions as well as the radius of the artificial boundary.     

Boundary Harnack Principle for fractional powers of Laplacian on the Sierpiński carpet

We prove the Boundary Harnack Principle related to fractional powers of Laplacian for some natural regions in the two-dimensional Sierpiński carpet. This is a natural application of some more general approach based on the Ikeda-Watanabe formula, that expresses the harmonic measure in terms the Green function of a given region and the Lévy measure of the semigroup.     

Comparison Principle for Viscosity Solutions with High Growth at Infinity

This paper is concerned with the comparison principle for viscosity solutions of the nonlinear elliptic equation F(Du, D²u} + |u|^{s-1}u =f in R^N, where f is uniformly continuous and F satisfies some conditions about p (p > 2}. We got the comparison principle for the viscosity solutions with some high growth at infinity, which relies on the relation between p and s.     

Boundary Harnack principle for subordinate Brownian motions

We establish a boundary Harnack principle for a large class of subordinate Brownian motions, including mixtures of symmetric stable processes, in κ$\kappa$-fat open sets (disconnected analogue of John domains). As an application of the boundary Harnack principle, we identify the Martin boundary and the minimal Martin boundary of bounded κ$\kappa$-fat open sets with respect to these processes with their Euclidean boundaries.     

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首先, 当$Q$是一个拟单调的q矩阵的时候, 我们找出最小的$Q$函数是一个Feller的转移函数的准则. 然后我们把这个结论应用于生成分支q矩阵并得到相应的生成分支过程的Feller准则. 特别地, 设$\theta$是分支q矩阵中的非线性数, 总是存在一个分点$\theta_0$满足$1\leq\theta_0\leq2$或$\theta_0<+\infty$使得 生成分支过程是否是Feller的要依据$\theta<\theta_0$或者$\theta>\theta_0$.     

Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in Feng and Wang (J. Appl. Probab. 44 938–949 2007) to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in Feng and Wang (J. Appl. Probab. 44 938–949 2007). To prove the main results, explicit Harnack inequality and super Poincaré inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.     

On the Principle of not Feeling the Boundary for Diffusion Processes

We derive the principle of not feeling the boundary for thetransition density function of a diffusion process from itsbasic short-time logarithmic asymptotic relation. This allowsus to extend this principle for more general diffusion processes.     

正则嵌入端与无穷远极大值原理

本文研究 Ｈ３（－１）中ＣＭＣ－ｌ曲面,对这类曲面的正则嵌入端进行了分类,分别称为第一类Ｃａｔｅｎｏｉｄ型的端,第二类Ｃａｔｅｎｏｉｄ型的端和Ｈｏｒｏｓｐｈｅｒｅ型的端,另外还讨论了局部的极大值原理和无穷远处的极大值原理．     

An Invariant Harnack Inequality for a Class of Hypoelliptic Ultraparabolic Equations

We show an invariant Harnack inequality for a class of hypoelliptic ultraparabolic operators with underlying homogeneous Lie group structures. As a byproduct we prove a Liouville type theorem for the related stationary operators. We also introduce a notion of link of homogeneous Lie Groups that allows us to show that our results apply to wide classes of operators.     

A Maximum Principle at Infinity for Surfaces with Constant Mean Curvature in Euclidean Space

Maximum principles at infinity generalize Hopf's maximum principle for hypersurfaces with constant mean curvature in R ⁿ . We establish such a maximum principle for parabolic surfaces in R³ with nonzero constant mean curvature and bounded Gaussian curvature.