Positive Harmonic Functions Research Articles

Invertibility Threshold for Nevanlinna Quotient Algebras

AbstractLet $\mathcal {N}$ be the Nevanlinna class, and let B be a Blaschke product. It is shown that the natural invertibility criterion in the quotient algebra $\mathcal {N} / B \mathcal {N}$ , that is, $|f| \ge e^{-H} $ on the set $B^{-1}\{0\}$ for some positive harmonic function H, holds if and only if the function $- \log |B|$ has a harmonic majorant on the set $\{z\in \mathbb {D}:\rho (z,\Lambda )\geq e^{-H(z)}\}$ , at least for large enough functions H. We also study the corresponding class of positive harmonic functions H on the unit disc such that the latter condition holds. We also discuss the analogous invertibility problem in quotients of the Smirnov class.

Positive harmonic functions on the Heisenberg group II

We present the classification of positive harmonic functions on the Heisenberg group in the case of the southwest measure.

On Subharmonic and Entire Functions of Small Order: After Kjellberg

Abstract We give a general method for constructing examples of transcendental entire functions of given small order, which allows precise control over the size and shape of the set where the minimum modulus of the function is relatively large. Our method involves a novel technique to obtain an upper bound for the growth of a positive harmonic function defined in a certain type of multiply connected domain, based on comparing the Harnack metric and hyperbolic metric, which gives a sharp estimate for the growth in many cases. Dedicated to the memory of Paddy Barry.

Discrete Harmonic Functions in the Three-Quarter Plane

In this article we are interested in finding positive discrete harmonic functions with Dirichlet conditions in three quadrants. Whereas planar lattice (random) walks in the quadrant have been well studied, the case of walks avoiding a quadrant has been developed lately. We extend the method in the quarter plane—resolution of a functional equation via boundary value problem using a conformal mapping—to the three-quarter plane applying the strategy of splitting the domain into two symmetric convex cones. We obtain a simple explicit expression for the algebraic generating function of harmonic functions associated to random walks avoiding a quadrant.

Discrete Harmonic Functions on Infinite Penny Graphs

In this paper, we study discrete harmonic functions on infinite penny graphs. For an infinite penny graph with bounded facial degree, we prove that the volume doubling property and the Poincar\'e inequality hold, which yields the Harnack inequality for positive harmonic functions. Moreover, we prove that the space of polynomial growth harmonic functions, or ancient solutions of the heat equation, with bounded growth rate has finite dimensional property.

Yau type gradient estimates for Δu + au(log⁡u)p + bu = 0 on Riemannian manifolds

In this paper, we consider the gradient estimates of the positive solutions to the following equation defined on a complete Riemannian manifold (M,g)Δu+au(log⁡u)p+bu=0, where a,b∈R and p is a rational number with p=k12k2+1≥2 where k1 and k2 are positive integer numbers. We obtain the gradient bound of a positive solution to the equation which does not depend on the bounds of the solution and the Laplacian of the distance function on (M,g). Our results can be viewed as a natural extension of Yau's estimates on positive harmonic function.

Positive harmonic functions on groups and covering spaces

We show 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 conjectured by Lyons and Sullivan in [14].

Schrödinger networks and their Cartesian products

A Schrödinger network is a suitable infinite graph on which certain potential‐theoretic aspects of the discrete Schrödinger equation can be studied. It is shown that the positive solutions of this discrete equation can be represented as integrals. The Cartesian product of Schrödinger networks, which has a bearing on Markov chains, is investigated. Also, we give a characterization of minimal positive harmonic functions on the the Cartesian product of Schrödinger networks.

Bôcher’s Theorem with Rough Coefficients

A classical theorem of Bocher says a positive harmonic function on a punctured domain $\mathcal {O}\setminus p$ in Euclidean space can be written as the sum of a constant multiple of the Green function with pole at p and a function harmonic on all of $\mathcal {O}$ . Here we establish such a result in the setting of functions on a Riemannian manifold with rather rough metric tensor.

Uniqueness results for positive harmonic functions on $$\overline{\mathbb {B}^{n}}$$ satisfying a nonlinear boundary condition

We prove some uniqueness results for positive harmonic functions on the unit ball satisfying a nonlinear boundary condition.

The growth rate of harmonic functions

We study the growth rate of harmonic functions in two aspects: gradient estimate and frequency. We obtain the sharp gradient estimate of positive harmonic function in geodesic ball of complete surface with nonnegative curvature. On complete Riemannian manifolds with non-negative Ricci curvature and maximal volume growth, further assume the dimension of the manifold is not less than three, we prove that quantitative strong unique continuation yields the existence of nonconstant polynomial growth harmonic functions. Also the uniform bound of frequency for linear growth harmonic functions on such manifolds is obtained, and this confirms a special case of Colding-Minicozzi conjecture on frequency.

An intuitive approach to the Martin boundary

An intuitive probabilistic alternative for the construction of the Martin boundary is presented along with a construction of maximal representing measures for positive harmonic functions.

From ℋ∞ to 𝒩. Pointwise properties and algebraic structure in the Nevanlinna class

Abstract This survey shows how, for the Nevanlinna class 𝒩 of the unit disc, one can define and often characterize the analogues of well-known objects and properties related to the algebra of bounded analytic functions ℋ∞: interpolating sequences, Corona theorem, sets of determination, stable rank, as well as the more recent notions of Weak Embedding Property and threshold of invertibility for quotient algebras. The general rule we observe is that a given result for ℋ∞ can be transposed to 𝒩 by replacing uniform bounds by a suitable control by positive harmonic functions. We show several instances where this rule applies, as well as some exceptions. We also briefly discuss the situation for the related Smirnov class.

Radial variation of Bloch functions on the unit ball of $\mathbb{R}^d$

In [9] Anderson’s conjecture was proven by comparing values of Bloch functions with the variation of the function. We extend that result on Bloch functions from two to arbitrary dimension and prove that \[ \int \limits_{[0, x]} \lvert \nabla b(\zeta) \rvert e^{b(\zeta)} \: d \lvert \zeta \rvert \lt \infty \; \textrm{.} \] In the second part of the paper, we show that the area or volume integral \[ \int \limits_{B^d} \lvert \nabla u(w) \rvert p(w,\theta) \: dA(w) \] for positive harmonic functions $u$ is bounded by the value $cu(0)$ for at least one $\theta$. The integral is also transferred to simply connected domains and interpreted from the point of view of stochastics. Several emerging open problems are presented.

Markov chains on {{mathbb {Z}}^+}: analysis of stationary measure via harmonic functions approach

We suggest a method for constructing a positive harmonic function for a wide class of transition kernels on {{mathbb {Z}}^+}. We also find natural conditions under which this harmonic function has a positive finite limit at infinity. Further, we apply our results on harmonic functions to asymptotically homogeneous Markov chains on {{mathbb {Z}}^+} with asymptotically negative drift which arise in various queueing models. More precisely, assuming that the Markov chain satisfies Cramér’s condition, we study the tail asymptotics of its stationary distribution. In particular, we clarify the impact of the rate of convergence of chain jumps towards the limiting distribution.

Alternative constructions of a harmonic function for a random walk in a cone

For a random walk killed at leaving a cone we suggest two new constructions of a positive harmonic function. These constructions allow one to remove a quite strong extendability assumption, which has been imposed in our previous paper (Denisov and Wachtel, 2015, Random walks in cones). As a consequence, all the limit results from that paper remain true for cones which are either convex or star-like and $C^{2}$.

Harnack and mean value inequalities on graphs

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.

Positive Harmonic Functions of Transformed Random Walks

In this paper, we will study the behavior of the space of positive harmonic functions associated with the random walk on a discrete group under the change of probability measure by a randomized stopping time. We show that this space remains unchanged after applying a bounded randomized stopping time.

Convexity of level lines of Martin functions and applications

Let $\Omega$ be an unbounded domain in $\mathbb{R}\times\mathbb{R}^{d}.$ A positive harmonic function $u$ on $\Omega$ that vanishes on the boundary of $\Omega$ is called a Martin function. In this note, we show that, when $\Omega$ is convex, the superlevel sets of a Martin function are also convex. As a consequence we obtain that if in addition $\Omega$ is symmetric, then the maximum of any Martin function along a slice $\Omega\cap (\{t\}\times\mathbb{R}^d)$ is attained at $(t,0).$

Series in a Lipschitz perturbation of the boundary for solving the Dirichlet problem

In a special Lipschitz domain treated as a perturbation of the upper half-space, we construct a perturbation theory series for a positive harmonic function with zero trace. The terms of the series are harmonic extensions to the half-space from its boundary of distributions defined by a recurrent formula and passage to the limit. The approximation error by a segment of the series is estimated via a power of the seminorm of the perturbation in the homogeneous Slobodestkiĭ space b 1−1/ . The series converges if the Lipschitz constant of the perturbation is small.