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A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

Qiang Tu, Xiaohuan Mu, Tiexin Guo, Goong Chen

Abstract

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of $L^{0}$-simplicial subdivisions of $L^{0}$-simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.

A new complete proof of the random Brouwer fixed point theorem and its implied consequences of unification

Abstract

We first establish a general random Sperner lemma by presenting a completely new approach for the theory of -simplicial subdivisions of -simplexes. Based on this, we are able to achieve a new complete proof of the random Brouwer fixed theorem in random Euclidean spaces, which can provide a solid foundation for various contemporary applications of interest. Afterward, we unify the works currently available and closely related to the random Brouwer fixed theorem: we first prove that the stochastic Brouwer fixed point theorem occurring elsewhere in stochastic analysis is equivalent to a special case of our random Brouwer fixed theorem, and then prove a general random Borsuk theorem and its equivalence with the random Brouwer fixed theorem. Finally, we conclude this paper with commentaries on recent state of study of the famous Schauder conjecture.

Paper Structure

This paper contains 9 sections, 26 theorems, 115 equations.

Table of Contents

  1. Introduction
  2. The random Brouwer fixed point theorem and a general random Sperner lemm. Preliminaries
  3. The random Brouwer fixed point theorem and its special cases
  4. A general $L^{0}$-simplicial subdivision and random Sperner lemma
  5. Proof of Theorem \ref{['thm.Sperner']}: Random Sperner Lemma
  6. Proof of Theorem \ref{['thm.Brouwer']}: The Random Brouwer Fixed Point Theorem
  7. The equivalence between Proposition \ref{['prop.2.3']} and Proposition \ref{['prop.2.4']}
  8. The equivalence between the random Brouwer fixed point theorem and random Borsuk theorem
  9. Commentaries on the state of the Schauder conjecture

Key Result

Theorem 2.1

Let $G$ be a $\sigma$-stable, a.s. bounded, $\mathcal{T}_{c}$-closed and $L^0$-convex subset of $L^0(\mathcal{F},\mathbb{R}^d)$ and $f: G \rightarrow G$ be a $\sigma$-stable $\mathcal{T}_{c}$-continuous mapping. Then $f$ has a fixed point.

Theorems & Definitions (61)

  • Theorem 2.1: The Random Brouwer Fixed Point Theorem
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • Definition 2.5: Guo2010GWXYC2025
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • Theorem 2.9: Random Sperner Lemma
  • Example 3.1
  • ...and 51 more