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Pré-Publication, Document De Travail Année : 2023

A remark on randomization of a general function of negative regularity

Tadahiro Oh
  • Fonction : Auteur
Mamoru Okamoto
  • Fonction : Auteur
Oana Pocovnicu
  • Fonction : Auteur

Résumé

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.

Dates et versions

ensl-04409994 , version 1 (22-01-2024)

Identifiants

Citer

Tadahiro Oh, Mamoru Okamoto, Oana Pocovnicu, Nikolay Tzvetkov. A remark on randomization of a general function of negative regularity. 2024. ⟨ensl-04409994⟩
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