A Measure-Theoretic Approach to Spontaneous Stochasticity

Abstract

Spontaneous stochasticity (SpSt), originating in Richardson's picture of turbulent dispersion and Lorenz's Eulerian view of finite-time loss of predictability, was later formulated under this name by Gaw\k{e}dzki and collaborators and developed in shell models by Mailybaev and collaborators. Whether it occurs in fully developed turbulence remains a major open question. Beyond a few specific classes of systems, however, SpSt has lacked a general mathematical definition. We introduce a measure-theoretic formalism in which it is understood as a measure-selection principle. Given an inviscid problem, a well-posed regularization, and an ambient measure, we study the pushforward of that measure by the regularized flow. Strong SpSt occurs when these pushforward measures converge to a non-Dirac probability law, replacing classical deterministic selection by statistical selection. For finite-dimensional systems, we establish several structural results. Our central attainability theorem shows that, whenever the inviscid problem is nonunique, any probability measure supported on the set of inviscid states can be selected as the limiting law of a suitable regularization. We also identify singular sets in the inviscid dynamics, detected through Dini-type directional growth, as necessary obstructions underlying nonuniqueness. We analyze the relation between SpSt and sensitivity to initial data, clarifying the scope and limitations of turbulence-inspired finite-time separation criteria. Finally, we develop a renormalization-(semi)group viewpoint in which limiting statistics arise as statistical attractors. Explicit examples illustrate how ambient measures, inviscid singularities, regularization scales, and initial-data sensitivity interact in the emergence of SpSt.