Positive Harmonic Functions Research Articles

Markov Chains in the Domain of Attraction of Brownian Motion in Cones

We consider a multidimensional Markov chain X converging to a multidimensional Brownian motion. We construct a positive harmonic function for X killed on exiting the cone. We show that its asymptotic behaviour is similar to that of the harmonic function of Brownian motion. We use the harmonic function to study the asymptotic behaviour of the tail distribution of the exit time τ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ au $$\\end{document} of X from a cone.

Every exponential group supports a positive harmonic function

We prove that all groups of exponential growth support non-constant positive harmonic functions. In fact, our results hold in the more general case of strongly connected, finitely supported Markov chains invariant under some transitive group of automorphisms for which the directed balls grow exponentially.

A proof of Guo-Wang’s conjecture on the uniqueness of positive harmonic functions in the unit ball

A proof of Guo-Wang’s conjecture on the uniqueness of positive harmonic functions in the unit ball

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.

Positive harmonic functions on biregular trees

We show that if f is a positive harmonic function on a biregular tree which has maximal growth along an infinite path in the tree, then every harmonic function g on the tree with 0 ≤ g ≤ f is a multiple of f, thus generalizing a result of Cartier about regular trees.

Harmonic functions for singular quadrant walks

We consider discrete (time and space) random walks confined to the quarter plane, with jumps only in directions (i,j) with i+j≥0 and small negative jumps, i.e., i,j≥−1. These walks are called singular, and were recently intensively studied from a combinatorial point of view. In this paper, we show how the compensation approach introduced in the 90ies by Adan, Wessels and Zijm may be applied to compute positive harmonic functions with Dirichlet boundary conditions. In particular, in case the random walks have a drift with positive coordinates, we derive an explicit formula for the escape probability, which is the probability to tend to infinity without reaching the boundary axes. These formulas typically involve famous recurrent sequences, such as the Fibonacci numbers. As a second step, we propose a probabilistic interpretation of the previously constructed harmonic functions and prove that they allow us to compute all positive harmonic functions of these singular walks. To that purpose, we derive the asymptotics of the Green functions in all directions of the quarter plane and use Martin boundary theory.

Mass, Capacitary Functions, and the Mass-to-Capacity Ratio

We study connections among the ADM mass, positive harmonic functions tending to zero at infinity, and the capacity of the boundary of asymptotically flat $3$-manifolds with nonnegative scalar curvature. First we give new formulae that detect the ADM mass via harmonic functions. Then we derive a family of monotone quantities and geometric inequalities if the underlying manifold has simple topology. As an immediate application, we observe several additional proofs of the $3$-dimensional Riemannian positive mass theorem. One proof leads to new, sufficient conditions that imply positivity of the mass via $C^0$-geometry of regions separating the boundary and $\infty$. A special case of such sufficient conditions shows, if a region enclosing the boundary has relative small volume, then the mass is positive. As further applications, we obtain integral identities for the mass-to-capacity ratio. We also promote the inequalities to become equality on spatial Schwarzschild manifolds outside rotationally symmetric spheres. Among other things, we show the mass-to-capacity ratio is always bounded below by one minus the square root of the normalized Willmore functional of the boundary. Prompted by our findings, we carry out a study of manifolds satisfying a constraint on the mass-to-capacity ratio. We point out such manifolds satisfy improved inequalities, their mass has an upper bound depending only on the boundary data, there are no closed minimal surfaces enclosing the boundary, and these manifolds include static extensions in the context of the Bartnik quasi-local mass.

The local uniform convergence of positive harmonic function sequence

The Harnack distance on space and its conformal invariance were constructed and studied by Herron. In this paper, we obtain the Harnack distance on domains in . Then, we use this concept to investigate some properties of the positive harmonic function class. These results are obtained in the complex plane, so it is advantageous to take some tools of the complex analysis. The main result of this paper is the property of the local uniform convergence of the positive harmonic sequences on a domain in the complex plane.

On global existence for semilinear wave equations with space-dependent critical damping

The global existence for semilinear wave equations with space-dependent critical damping $\partial_{t}^{2} u - \Delta u + \frac{V_0}{|x|} \partial_{t} u = f(u)$ in an exterior domain is dealt with, where $f(u) = |u|^{p-1} u$ and $f(u) = |u|^{p}$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata–Todorova–Yordanov [J. Math. Soc. Japan (2013), 183–236] but the argument in this paper clarifies the precise dependence of the location of the support of initial data. The blowup phenomena are verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.

Monotonicity Formulas for Harmonic Functions in textrm{RCD}(0,N) Spaces

We generalize to the textrm{RCD}(0,N) setting a family of monotonicity formulas by Colding and Minicozzi for positive harmonic functions in Riemannian manifolds with nonnegative Ricci curvature. Rigidity and almost rigidity statements are also proven, the second appearing to be new even in the smooth setting. Motivated by the recent work in Agostiniani et al. (Invent. Math. 222(3):1033–1101, 2020), we also introduce the notion of electrostatic potential in textrm{RCD} spaces, which also satisfies our monotonicity formulas. Our arguments are mainly based on new estimates for harmonic functions in textrm{RCD}(K,N) spaces and on a new functional version of the ‘(almost) outer volume cone implies (almost) outer metric cone’ theorem.

Liouville type theorems for positive harmonic functions on the unit ball with a nonlinear boundary condition

Liouville type theorems for positive harmonic functions on the unit ball with a nonlinear boundary condition

Isolated Singularities of Positive Harmonic Functions in Bounded Domains

The singularity of positive harmonic functions has been studied in the punctured unit open ball in B(0,1)/{0} [1]. The representation of a positive harmonic function in B(0,1)/{0} is given by Bôcher’s Theorem. In this work, the representation of Bôchre’s has been generalized to a punctured bounded domain in R n Omiga/{Xo} . Then the work has been extended to represent positive harmonic functions with two isolated singularities in a bounded domain of R n .

Enumeration of three-quadrant walks via invariants: some diagonally symmetric models

AbstractIn the past $20$ years, the enumeration of plane lattice walks confined to a convex cone—normalized into the first quadrant—has received a lot of attention, stimulated the development of several original approaches, and led to a rich collection of results. Most of these results deal with the nature of the associated generating function: for which models is it algebraic, D-finite, D-algebraic? By model, what we mean is a finite collection of allowed steps.More recently, similar questions have been raised for nonconvex cones, typically the three-quadrant cone $\mathcal {C} = \{ (i,j) : i \geq 0 \text { or } j \geq 0 \}$ . They turn out to be more difficult than their quadrant counterparts. In this paper, we investigate a collection of eight models in $\mathcal {C}$ , which can be seen as the first level of difficulty beyond quadrant problems. This collection consists of diagonally symmetric models in $\{-1, 0,1\}^2\setminus \{(-1,1), (1,-1)\}$ . Three of them are known not to be D-algebraic. We show that the remaining five can be solved in a uniform fashion using Tutte’s notion of invariants, which has already proved useful for some quadrant models. Three models are found to be algebraic, one is (only) D-finite, and the last one is (only) D-algebraic. We also solve in the same fashion the diagonal model $\{ \nearrow , \nwarrow , \swarrow , \searrow \}$ , which is D-finite. The three algebraic models are those of the Kreweras trilogy, $\mathcal S=\{\nearrow , \leftarrow , \downarrow \}$ , $\mathcal S^*=\{\rightarrow , \uparrow , \swarrow \}$ , and $\mathcal S\cup \mathcal S^*$ .Our solutions take similar forms for all six models. Roughly speaking, the square of the generating function of three-quadrant walks with steps in $\mathcal S$ is an explicit rational function in the quadrant generating function with steps in $\mathscr S:= \{(j-i,j): (i,j) \in \mathcal S\}$ . We derive various exact or asymptotic corollaries, including an explicit algebraic description of a positive harmonic function in $\mathcal C$ for the (reverses of the) five models that are at least D-finite.

The Rigidity and Stability of Gradient Estimates

In this note, we obtain the rigidity of the sharp Cheng-Yau gradient estimate for positive harmonic functions on surfaces with non-negative Gaussian curvature, the rigidity of the sharp Li-Yau gradient estimate for positive solutions to heat equations and the related estimates for Dirichlet Green’s functions on Riemannian manifolds with non-negative Ricci curvature. Moreover, we also obtain the corresponding stability results.

On the pointwise converse of Fatou’s theorem for Euclidean and real hyperbolic spaces

In this article, we extend a result of L. Loomis and W. Rudin, regarding boundary behavior of positive harmonic functions on the upper half space ℝ n+1+ . We show that similar results remain valid for more general approximate identities. We apply this result to prove a result regarding boundary behavior of certain nonnegative eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn. We shall also prove a generalization of a result regarding large time behavior of a solution of the heat equation proved in [17]. We use this result to prove a result regarding asymptotic behavior of certain eigenfunctions of the Laplace-Beltrami operator on real hyperbolic space ℍn.

On the Strong Liouville Property of Covering Spaces

Lyons and Sullivan conjectured in Lyons and Sullivan (J. Differential Geom. 19(2), 299–323, 1984) that if p : M → N is a normal Riemannian covering, with N closed, and M has exponential volume growth, then there are non-constant, positive harmonic functions on M. This was proved recently in Polymerakis (Adv. Math. 379, 107552–107558, 2021) exploiting the Lyons-Sullivan discretization and some sophisticated estimates on the green metric on groups. In this note, we provide a self-contained proof relying only on elementary properties of the Brownian motion.

Growth of harmonic functions on biregular trees

On a biregular tree of degrees $q+1$ and $r+1$, we study the growth of two classes of harmonic functions. First, we prove that if $f$ is a bounded harmonic function on the tree and $x$, $y$ are two adjacent vertices, then $|f(x)-f(y)|\leq 2 (qr-1)\|f\|_\infty/((q+1)(r+1))$, thus generalizing a result of Cohen and Colonna for regular trees. Next, we prove that if $f$ is a positive harmonic function on the tree and $x$, $y$ are two vertices with $d(x,y)=2$, then $f(x)/(qr)\leq f(y)\leq qr\cdot f(x)$.

Constructing discrete harmonic functions in wedges

We propose a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, we prove that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function.

Fatou's theorem for non-commutative measures

A classical theorem of Fatou asserts that the Radon–Nikodym derivative of any finite and positive Borel measure, μ, with respect to Lebesgue measure on the complex unit circle, is recovered as the non-tangential limits of its Poisson transform in the complex unit disk. This positive harmonic Poisson transform is the real part of an analytic function whose Taylor coefficients are in fixed proportion to the conjugate moments of μ.Replacing Taylor series in one variable by power series in several non-commuting variables, we show that Fatou's Theorem and related results have natural extensions to the setting of positive harmonic functions in an open unit ball of several non-commuting matrix-variables, and a corresponding class of positive non-commutative (NC) measures. Here, an NC measure is any positive linear functional on a certain self-adjoint unital subspace of the Cuntz–Toeplitz algebra, the C⁎−algebra generated by the left creation operators on the full Fock space.

Potential Theory on Gromov Hyperbolic Spaces

Abstract Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal hypersurfaces. Results include boundary Harnack inequalities and a complete classification of positive harmonic functions in terms of the Martin boundary which is identified with the geometric Gromov boundary.