On the Stochastic Burgers Equation and the Axiom of Choice

TL;DR

This paper explores the connection between the stochastic Burgers equation and the Axiom of Choice, analyzing how different mathematical assumptions influence the existence of solutions and invariant measures.

Contribution

It compares solution constructions with and without Tychonov compactness, shedding light on the role of the Axiom of Choice in stochastic PDEs.

Findings

Tychonov compactness is crucial for certain solution existence proofs.

The representation of solutions affects the applicability of the Axiom of Choice.

Insights into the foundational assumptions underlying stochastic PDE solutions.

Abstract

The equivalence of the Choice Axiom and Tychonov compactness was proved by Kelley in 1950. Tychonov compactness is required to prove existence of the minimiser of an action functional under standard hypotheses. Solutions to the inviscid Burgers equation may be constructed by considering the minimising trajectories of associated action functionals. This construction has been used in the literature to construct an invariant measure for the stochastic inviscid Burgers equation. This article compares results that may be obtained with and without using this representation and draws conclusions about the Choice Axiom.