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Heat kernel and Green function estimates on affine buildings

Bartosz Trojan

Abstract

We obtain the optimal global upper and lower bounds for the transition density $p_n(x,y)$ of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function.

Heat kernel and Green function estimates on affine buildings

Abstract

We obtain the optimal global upper and lower bounds for the transition density of a finite range isotropic random walk on affine buildings. We present also sharp estimates for the corresponding Green function.

Paper Structure

This paper contains 15 sections, 17 theorems, 431 equations.

Table of Contents

  1. Introduction
  2. Organization of the paper
  3. Combinatorial and analytic preliminaries
  4. Convex combinations of exponentials and the function s
  5. Analytic lemmas about multiple derivation
  6. Variation on the marriage lemma
  7. Affine buildings
  8. Root systems, weights and coweigths
  9. Building, thicknesses and (co)type
  10. Spherical harmonic analysis
  11. Asymptotics
  12. Random walks
  13. Heat kernels
  14. Green functions
  15. Asymptotic in the exceptional case

Key Result

Theorem 1

Let $(\omega_n : n \in \mathbb{N})$ be a sequence of co-weights such that the sphere centered at $O$ and radius $\omega_n$ is contained in the support of $p(n; O, \cdot\,)$. Suppose that where $\delta_n = n^{-1} \omega_n$. Then for any sequence of good vertices $(x_n : n \in \mathbb{N})$ such that the Weyl distance between $O$ and $x_n$ equals $\omega_n$, we have The constant $C_0$ is absolute.

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Lemma 1: Faà di Bruno's formula
  • Lemma 2
  • proof
  • Lemma 3
  • ...and 26 more