Spontaneously stochastic Arnold's cat

TL;DR

This paper introduces a rigorous model demonstrating how tiny noise in a deterministic turbulence system can produce large-scale stochastic behavior, shedding light on fundamental turbulence phenomena and opening avenues for experimental validation.

Contribution

It provides a simple, rigorously solved model that explains spontaneous stochasticity in turbulence and connects to key open problems in the mathematical theory of turbulence.

Findings

Small-scale noise induces large-scale stochasticity.

Model exhibits non-uniqueness of solutions and wild weak solutions.

Suggests new experimental and application pathways.

Abstract

We propose a simple model for the phenomenon of Eulerian spontaneous stochasticity in turbulence. This model is solved rigorously, proving that infinitesimal small-scale noise in otherwise a deterministic multi-scale system yields a large-scale stochastic process with Markovian properties. Our model shares intriguing properties with open problems of modern mathematical theory of turbulence, like non-uniqueness of the inviscid limit, existence of wild weak solutions and explosive effect of random perturbations. Thereby, it proposes rigorous, often counterintuitive answers to these questions. Besides its theoretical value, our model opens new ways for the experimental verification of spontaneous stochasticity, and suggests new applications beyond fluid dynamics.