A nonlinear Strassen law for singular SPDEs

Abstract

A result of Arcones implies that if a measure-preserving linear operator $S$ on an abstract Wiener space $(X,H,\mu )$ is strongly mixing, then the set of limit points of the random sequence $((2\log n{)}^{-1/2}{S}^n(x){)}_{n\in \mathbb{N}}$ equals the unit ball of $H$ for a.e. $x\in X$, which may be seen as a generalization of the classical Strassen's law of the iterated logarithm. We extend this result to the case of a continuous parameter $n$ and higher Gaussian chaoses, and we also prove a contraction-type principle for Strassen laws of such chaoses. We then use these extensions to recover or prove Strassen-type laws for a broad collection of processes derived from a Gaussian measure, including "nonlinear" Strassen laws for singular SPDEs such as the KPZ equation.

3rd version: fixed various small typos appearing in the published version

References in corpus (1)

• Characterizing models in regularity structures: a quasilinear case