A nonlinear Strassen law for singular SPDEs

TL;DR

This paper extends Strassen's law of the iterated logarithm to nonlinear and continuous settings, applying it to Gaussian chaoses and singular SPDEs like the KPZ equation, revealing new limit behaviors.

Contribution

It generalizes Strassen's law to continuous parameters, higher Gaussian chaoses, and nonlinear SPDEs, introducing a contraction principle for these laws.

Findings

Extended Strassen law to continuous parameters and Gaussian chaoses.

Proved a contraction principle for Strassen laws of chaoses.

Established nonlinear Strassen laws for singular SPDEs such as KPZ.

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.