A nonlinear Strassen Law for singular SPDEs

Abstract

A result of Arcones [Arc95] implies that if a measure-preserving linear operator S on an abstract Wiener space $(X,H,\mathrm{\mu })$ is strongly mixing, then the set of limit points of the random sequence ${({(2logn)}^{-1∕2}{S}^{n}x)}_{n\in \mathbb{N}}$ equals the unit ball of H for μ-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.