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The mean value property (corrected version)

Vilmos Totik

TL;DR

The paper analyzes discrete and continuous mean value properties and proves that boundedness or positivity forces constancy for functions satisfying these averaging conditions, across lattice and plane settings. It develops elementary tools—a discrete maximum principle, an iterative fixed-point scheme, and extremal-argument techniques—to connect discrete potential theory, random walks, Brownian motion, the Dirichlet problem, and minimal surfaces. By framing these results within a convex-analytic viewpoint via Krein–Milman (and Choquet–Deny) theory, it reveals a unifying structure behind local averaging constraints. The work thus bridges probabilistic, analytic, and geometric perspectives, with implications for Dirichlet problems and harmonic measures in multiple dimensions.

Abstract

The paper deals with some elementary problems about various mean value properties and their connections to harmonic functions and random walks.

The mean value property (corrected version)

TL;DR

The paper analyzes discrete and continuous mean value properties and proves that boundedness or positivity forces constancy for functions satisfying these averaging conditions, across lattice and plane settings. It develops elementary tools—a discrete maximum principle, an iterative fixed-point scheme, and extremal-argument techniques—to connect discrete potential theory, random walks, Brownian motion, the Dirichlet problem, and minimal surfaces. By framing these results within a convex-analytic viewpoint via Krein–Milman (and Choquet–Deny) theory, it reveals a unifying structure behind local averaging constraints. The work thus bridges probabilistic, analytic, and geometric perspectives, with implications for Dirichlet problems and harmonic measures in multiple dimensions.

Abstract

The paper deals with some elementary problems about various mean value properties and their connections to harmonic functions and random walks.

Paper Structure

This paper contains 11 sections, 3 theorems, 25 equations, 5 figures.

Table of Contents

  1. Some problem challenges
  2. The maximum principle
  3. Random walks
  4. An iteration process
  5. Solution to Problem 1
  6. Sketch of the solution to Problem 2
  7. The continuous mean value property and harmonic functions
  8. The Dirichlet problem and soap films
  9. Discretization and Brownian motions
  10. Solution to Problem 4
  11. The Krein-Milman theorem

Key Result

Theorem 1

(Minimum/maximum principle) If a function with the discrete mean value property on the integer lattice attains somewhere its smallest/largest value, then it must be constant.

Figures (5)

  • Figure 1:
  • Figure 2: A sample random walk starting at $P=A$ and terminating at $Z$: ABCDADABEZ
  • Figure 3: The plane curve $\gamma$ and its lift-up $\Gamma$
  • Figure 4: The domain $\Omega$ enclosed by the closed curve and the region $G$ of squares lying inside $\Omega$ (with darker shaded boundary squares)
  • Figure 5: A Brownian motion

Theorems & Definitions (3)

  • Theorem 1
  • Theorem 2
  • Theorem 3