February 2024 Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions
Antonio Agresti, Mark Veraar
Author Affiliations +
Ann. Inst. H. Poincaré Probab. Statist. 60(1): 413-430 (February 2024). DOI: 10.1214/22-AIHP1333


In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω), and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove Lp((0,T)×Ω,tκdt;Hσ,q(Td))-estimates. The main novelty is that we do not require p=q. Moreover, we allow arbitrary σR and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.

Dans cet article, nous considérons une EDPS où le terme dominant est un opérateur du second ordre avec des conditions aux limites périodiques, des coefficients mesurables en (t,ω) et de regularité Hölderienne en espace. En supposant des conditions de parabolicité stochastique, nous prouvons des estimations de type Lp((0,T)×Ω,tκdt;Hσ,q(Td)). La principale nouveauté est que nous n’avons pas besoin de p=q. De plus, nous autorisons σR à être arbitraire et des poids en temps. De telles estimations de régularité mixtes jouent un rôle crucial dans les applications aux EDPS non linéaires, ce qui ressort clairement de nos travaux précédents. Pour prouver nos principaux résultats, nous développons une méthode générale perturbative pour les EDPS. De plus, nous prouvons un nouveau résultat sur la multiplication ponctuelle dans des espaces à régularité fractionnaire.

Funding Statement

The first author has been partially supported by the Nachwuchsring – Network for the promotion of young scientists – at TU Kaiserslautern. The second author is supported by the VIDI subsidy 639.032.427 of the Netherlands Organisation for Scientific Research (NWO).


The authors thank the anonymous referees and Max Sauerbrey for careful reading and helpful suggestions.


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Antonio Agresti. Mark Veraar. "Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions." Ann. Inst. H. Poincaré Probab. Statist. 60 (1) 413 - 430, February 2024. https://doi.org/10.1214/22-AIHP1333


Received: 3 June 2021; Revised: 28 July 2022; Accepted: 3 October 2022; Published: February 2024
First available in Project Euclid: 3 March 2024

MathSciNet: MR4718386
Digital Object Identifier: 10.1214/22-AIHP1333

Primary: 60H15
Secondary: 35B65 , 42B37 , 46F10 , 47D06 , 60H15

Keywords: periodic boundary conditions , Perturbation theory , pointwise multipliers , Second order operators , stochastic evolution equations , stochastic maximal regularity

Rights: Copyright © 2024 Association des Publications de l’Institut Henri Poincaré


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Vol.60 • No. 1 • February 2024
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