Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence

2020/01/02 by Antonio Agresti, Mark Veraar, Agresti, Antonio +1 · 13 citations
Economics, Econometrics and Finance · Mathematics · #35K59 #35K90 #35R60 #42B37 #47D06 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Primary: 60H15 #Probability (math.PR) #Secondary: 35B65 #Stochastic processes and financial applications #Stochastic processes and statistical mechanics #math.AP #math.FA #math.PR #msc:35B65 #msc:35K59 #msc:35K90 #msc:35R60 #msc:42B37 #msc:47D06 #msc:60H15

paper · pdf · doi:10.48550/arxiv.2001.00512

Accepted for publication in Nonlinearity. Few typos corrected

openalex publication_date 2020/01/02 · arxiv created 2022/04/29 · arxiv updated 2022/05/02 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an L^p-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burger's equation, the Allen-Cahn equation, the Cahn-Hilliard equation, reaction-diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss-Simonett-Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.

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