Nonnegative Harmonic Functions Research Articles

An Explicit Form of Harnack’s Inequality for Poisson’s Dirichlet Problem

The explicit form of Harnack’s inequality for non-negative harmonic functions in the open ball plays an important role in harmonic analysis and elliptic partial differential equations. In this paper, we establish the explicit form of Harnack’s inequality for Poisson’s Dirichlet problem in the open ball with non-negative boundary data. We involve the Green’s function for Laplacian operator to find it out. Harnack’s inequality for positive harmonic functions can be followed from Harnack’s inequality for Poisson’s Dirichlet problem when the source function is set to be zero.

Elliptic Harnack inequality for ℤd

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.

On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid.

Discrete harmonic functions in Lipschitz domains

We prove the existence and uniqueness of a discrete nonnegative harmonic function for a random walk satisfying finite range, centering and ellipticity conditions, killed when leaving a globally Lipschitz domain in $\mathbb{Z} ^{d}$. Our method is based on a systematic use of comparison arguments and discrete potential-theoretical techniques.

Potential theory of subordinate killed Brownian motion

Let $W^D$ be a killed Brownian motion in a domain $D\subset {\mathbb R}^d$ and $S$ an independent subordinator with Laplace exponent $\phi$. The process $Y^D$ defined by $Y^D_t=W^D_{S_t}$ is called a subordinate killed Brownian motion. It is a Hunt process with infinitesimal generator $-\phi(-\Delta|_D)$, where $\Delta|_D$ is the Dirichlet Laplacian. In this paper we study the potential theory of $Y^D$ under a weak scaling condition on the derivative of $\phi$. We first show that non-negative harmonic functions of $Y^D$ satisfy the scale invariant Harnack inequality. Subsequently we prove two types of scale invariant boundary Harnack principles with explicit decay rates for non-negative harmonic functions of $Y^D$. The first boundary Harnack principle deals with a $C^{1,1}$ domain $D$ and non-negative functions which are harmonic near the boundary of $D$, while the second one is for a more general domain $D$ and non-negative functions which are harmonic near the boundary of an interior open subset of $D$. The obtained decay rates are not the same, reflecting different boundary and interior behaviors of $Y^D$.

Harnack Inequality for Subordinate Random Walks

In this paper, we consider a large class of subordinate random walks X on the integer lattice $$\mathbb {Z}^d$$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying some mild scaling conditions at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for nonnegative harmonic functions.

Doob–Martin boundary of Rémy’s tree growth chain

Author(s): Evans, SN; Grubel, R; Wakolbinger, A | Abstract: Remy's algorithm is a Markov chain that iteratively generates a sequence of random trees in such a way that the nth tree is uniformly distributed over the set of rooted, planar, binary trees with 2n + 1 vertices. We obtain a concrete characterization of the Doob-Martin boundary of this transient Markov chain and thereby delineate all the ways in which, loosely speaking, this process can be conditioned to go to infinity at large times. A (deterministic) sequence of finite rooted, planar, binary trees converges to a point in the boundary if for each m the random rooted, planar, binary tree spanned by m + 1 leaves chosen uniformly at random from the nth tree in the sequence converges in distribution as n tends to infinity-a notion of convergence that is analogous to one that appears in the recently developed theory of graph limits. We show that a point in the Doob-Martin boundary may be identified with the following ensemble of objects: a complete separable ℝ-tree that is rooted and binary in a suitable sense, a diffuse probability measure on the R-tree that allows us to make sense of sampling points from it, and a kernel on the R-tree that describes the probability that the first of a given pair of points is below and to the left of their most recent common ancestor while the second is below and to the right. Two such ensembles represent the same point in the boundary if for each m the random, rooted, planar, binary trees spanned by m + 1 independent points chosen according to the respective probability measures have the same distribution. Also, the Doob-Martin boundary corresponds bijectively to the set of extreme point of the closed convex set of nonnegative harmonic functions that take the value 1 at the binary tree with 3 vertices; in other words, the minimal and full Doob-Martin boundaries coincide. These results are in the spirit of the identification of graphons as limit objects in the theory of graph limits.

Boundary Harnack Principle and Gradient Estimates for Fractional Laplacian Perturbed by Non-local Operators

Suppose $d\ge 2$ and $0<\beta<\alpha<2$. We consider the non-local operator $\mathcal{L}^{b}=\Delta^{\alpha/2}+\mathcal{S}^{b}$, where $$\mathcal{S}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right)\frac{b(x,z)}{|z|^{d+\beta}}\,dy.$$ Here $b(x,z)$ is a bounded measurable function on $\mathbb{R}^{d}\times\mathbb{R}^{d}$ that is symmetric in $z$, and $\mathcal{A}(d,-\beta)$ is a normalizing constant so that when $b(x, z)\equiv 1$, $\mathcal{S}^{b}$ becomes the fractional Laplacian $\Delta^{\beta/2}:=-(-\Delta)^{\beta/2}$. In other words, $$\mathcal{L}^{b}f(x):=\lim_{\varepsilon\to 0}\mathcal{A}(d,-\beta)\int_{|z|>\varepsilon}\left(f(x+z)-f(x)\right) j^b(x, z)\,dz,$$ where $j^b(x, z):= \mathcal{A}(d,-\alpha) |z|^{-(d+\alpha)}+ \mathcal{A}(d,-\beta) b(x, z)|z|^{-(d+\beta)}$. It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when $j^b(x, z)\geq 0$ on $\mathbb{R}^d\times \mathbb{R}^d$, there is a conservative Feller process $X^{b}$ having $\mathcal{L}^b$ as its infinitesimal generator. In this paper we establish, under certain conditions on $b$, a uniform boundary Harnack principle for harmonic functions of $X^b$ (or equivalently, of $\mathcal{L}^b$) in any $\kappa$-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of $X^{b}$ in open sets.

Fatou’s theorem for subordinate Brownian motions with Gaussian components on $C^{1,1}$ open sets

We prove Fatou’s theorem for nonnegative harmonic functions with respect to killed subordinate Brownian motions with Gaussian components on bounded $C^{1,1}$ open sets $D$. We prove that nonnegative harmonic functions with respect to such processes on $D$ converge nontangentially almost everywhere with respect to the surface measure as well as the harmonic measure restricted to the boundary of the domain. In order to prove this, we first prove that the harmonic measure restricted to $\partial D$ is mutually absolutely continuous with respect to the surface measure. We also show that tangential convergence fails on the unit ball.

Hausdorff Measure Estimates and Lipschitz Regularity in Inhomogeneous Nonlinear Free Boundary Problems

In this paper, we prove a Hausdorff measure estimate for the free boundaries of subsolutions of fully nonlinear and quasilinear equations of the type \({F(D^2u,x)\geqq f(x)}\) and \({{\rm div}\,A(x,\nabla u)\geqq \mu}\) where \({f \in L^{q}, q >N}\) and μ is a signed Radon measure with some appropriate growth condition. Gradient estimates for nonnegative harmonic functions with bounded normal derivatives along the boundary obtained by Caffarelli and Salsa (Geometric Approach to Free Boundary Problems, 2005) are extended to the context of inhomogeneous problems involving fully nonlinear and p-Laplace equations. As an application, Lipschitz regularity is obtained for one phase solutions of inhomogeneous nonlinear free boundary problems.

Harmonic functions on the branching graph associated with the infinite wreath product of a compact group

A detailed study of the characters of S∞(T), the wreath product of compact group T with the infinite symmetric group S∞, is indispensable for harmonic analysis on this big group. In preceding works, we investigated limiting behavior of characters of the finite wreath product Sn(T) as n→∞ and its connection with characters of S∞(T). This paper takes a dual approach to these problems. We study harmonic functions on Y(T^), the branching graph of the inductive system of Sn(T)’s, and give a classification of the minimal nonnegative harmonic functions on it. This immediately implies a classification of the characters of S∞(T), which is a logically independent proof of the one obtained in earlier works. We obtain explicit formulas for minimal nonnegative harmonic functions on Y(T^) and Martin integral expressions for harmonic functions.

On harmonic functions of killed random walks in convex cones

We prove the existence of uncountably many nonnegative harmonic functions for random walks in the euclidean space with non-zero drift, killed when leaving general convex cones with vertex in 0. We also make the natural conjecture about the Martin boundary for lattice random walks in general convex cones in two dimensions. Proving that the set of harmonic functions found is the full Martin boundary for these processes is an open problem.

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C1,1 open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C1,1 open set D and identify the Martin boundary of D with respect to the subordinate Brownian motion with the Euclidean boundary.

Oscillation of harmonic functions for subordinate Brownian motion and its applications

In this paper, we establish an oscillation estimate of nonnegative harmonic functions for a pure-jump subordinate Brownian motion. The infinitesimal generator of such subordinate Brownian motion is an integro-differential operator. As an application, we give a probabilistic proof of the following form of relative Fatou theorem for such subordinate Brownian motion X in a bounded κ-fat open set; if u is a positive harmonic function with respect to X in a bounded κ-fat open set D and h is a positive harmonic function in D vanishing on Dc, then the non-tangential limit of u/h exists almost everywhere with respect to the Martin-representing measure of h.

Boundary Multifractal Behaviour for Harmonic Functions in the Ball

It is well known that if h is a nonnegative harmonic function in the ball of \(\mathbb R^{d+1}\) or if h is harmonic in the ball with integrable boundary values, then the radial limit of h exists at almost every point of the boundary. In this paper, we are interested in the exceptional set of points of divergence and in the speed of divergence at these points. In particular, we prove that for generic harmonic functions and for any β ∈ [0,d], the Hausdorff dimension of the set of points ξ on the sphere such that h(rξ) looks like (1 − r) − β is equal to d − β.

Constructing a harmonic function for an irreducible nonnegative matrix with convergence parameter R >1

Let Q be a nonnegative, irreducible matrix with radius of convergence R>1. Suppose that J is a substochastic Toeplitz transition kernel J that corresponds to a random walk that is killed on the negative integers but has a positive probability of escaping to infinity. Results by Doney, J. London Math. Soc. 58 (1998) 761–768, show J has a nonnegative harmonic function U(i), i∈ℕ that converges to a constant as i→∞. We show that Q has the same property, provided that Q and J are close. Note that Q is not required to be substochastic. The results in this paper are useful in understanding large deviations of a node in a queueing network that occur because of ‘cascades’, that is, because of earlier large deviations in some other part of the network that then spill over and cascade into the node of interest.

Harnack Inequality and Regularity for a Product of Symmetric Stable Process and Brownian Motion

In this paper, we consider a product of a symmetric stable process in ℝd and a one-dimensional Brownian motion in ℝ + . Then we define a class of harmonic functions with respect to this product process. We show that bounded non-negative harmonic functions in the upper-half space satisfy Harnack inequality and prove that they are locally Hölder continuous. We also argue a result on Littlewood–Paley functions which are obtained by the α-harmonic extension of an L p(ℝd) function.

Representing measures in potential theory and an ideal boundary

A primary motivating application of the author’s work on measure theory was a nonstandard construction of standard representing measures for positive harmonic functions. That work yielded new standard weak convergence methods for constructing such measures on spaces of extreme harmonic functions in very general settings. The search for a Martin-type ideal boundary for the placement of those measures resulted in a new almost everywhere regular boundary that supported the representing measures for a large proper subclass of all nonnegative harmonic functions. In this note, we outline the construction of the rich measure spaces that are now called Loeb measure spaces in the literature. We then review the application of these measure spaces to the construction of representing measures. We finish with the problem of constructing an appropriate boundary associated with the nonstandard construction of general representing measures that supports all of those measures.

Harmonic Functions for a Class of Integro-differential Operators

We consider the operator \({\cal L}\) defined on \(C^2({\Bbb R}^d)\) functions by $$ \begin{array}{lll} {\cal L} f(x)&=&\frac{1}{2}\sum\limits_{i,j=1}^d a_{ij}(x)\frac{\partial^2f(x)}{\partial x_i\partial x_j}+\sum\limits_{i=1}^d b_i(x)\frac{\partial f(x)}{\partial x_i}\\ &&+\int_{{\Bbb R}^d\backslash\{0\}}\left[f(x+h)-f(x)-1_{(|h|\leq1)}h\cdot {\nabla} f(x)\right]n(x,h)dh. \end{array} $$ Under the assumption that the local part of the operator is uniformly elliptic and with suitable conditions on n(x,h), we establish a Harnack inequality for functions that are nonnegative in \( {\Bbb R}^d \) and harmonic in a domain. We also show that the Harnack inequality can fail without suitable conditions on n(x,h). A regularity theorem for those nonnegative harmonic functions is also proved.

Estimates and structure of α-harmonic functions

We prove a uniform boundary Harnack inequality for nonnegative harmonic functions of the fractional Laplacian on arbitrary open set D. This yields a unique representation of such functions as integrals against measures on D c∪ {∞} satisfying an integrability condition. The corresponding Martin boundary of D is a subset of the Euclidean boundary determined by an integral test.