A remark on randomization of a general function of negative regularity

TL;DR

This paper investigates the limitations of constructing Wick powers for randomized initial data and stochastic PDEs with negative regularity, highlighting the necessity of additional Fourier-Lebesgue regularity for well-posedness.

Contribution

It demonstrates the failure of probabilistic well-posedness for nonlinear wave equations with general randomized initial data in negative Sobolev spaces, emphasizing the need for Fourier-Lebesgue regularity.

Findings

Wick powers do not exist for functions outside $L^2( imes)$

Probabilistic well-posedness fails in negative Sobolev spaces

Fourier-Lebesgue $ ext{ extgamma}$-radonifying regularity is necessary

Abstract

In the study of partial differential equations (PDEs) with random initial data and singular stochastic PDEs with random forcing, we typically decompose a classically ill-defined solution map into two steps, where, in the first step, we use stochastic analysis to construct various stochastic objects. The simplest kind of such stochastic objects is the Wick powers of a basic stochastic term (namely a random linear solution, a stochastic convolution, or their sum). In the case of randomized initial data of a general function of negative regularity for studying nonlinear wave equations (NLW), we show necessity of imposing additional Fourier-Lebesgue regularity for constructing Wick powers by exhibiting examples of functions slightly outside $L^2(\mathbb T^d)$ such that the associated Wick powers do not exist. This shows that probabilistic well-posedness theory for NLW with general randomized initial data fails in negative Sobolev spaces (even with renormalization). Similar examples also apply to stochastic NLW and stochastic nonlinear heat equations with general white-in-time stochastic forcing, showing necessity of appropriate Fourier-Lebesgue $\gamma$-radonifying regularity in the construction of the Wick powers of the associated stochastic convolution.