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

TL;DR

This paper introduces a novel approach to prove the random Brouwer fixed point theorem using a new random Sperner lemma, unifies related stochastic fixed point theorems, and discusses implications for the Schauder conjecture.

Contribution

It provides a new complete proof of the random Brouwer fixed point theorem and unifies various related stochastic fixed point results.

Findings

Established a new random Sperner lemma for $L^{0}$-simplicial subdivisions.

Proved the equivalence of stochastic Brouwer fixed point theorem with the new random version.

Connected the random Brouwer fixed theorem to the random Borsuk theorem and discussed the Schauder conjecture.

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.