November 2022 Optimal regularity in time and space for stochastic porous medium equations
Stefano Bruno, Benjamin Gess, Hendrik Weber
Author Affiliations +
Ann. Probab. 50(6): 2288-2343 (November 2022). DOI: 10.1214/22-AOP1583


We prove optimal regularity estimates in Sobolev spaces in time and space for solutions to stochastic porous medium equations. The noise term considered here is multiplicative, white in time and coloured in space. The coefficients are assumed to be Hölder continuous, and the cases of smooth coefficients of, at most, linear growth as well as u are covered by our assumptions. The regularity obtained is consistent with the optimal regularity derived for the deterministic porous medium equation in (J. Eur. Math. Soc. 23 (2021) 425–465, Anal. PDE 13 (2020) 2441–2480) and the presence of the temporal white noise. The proof relies on a significant adaptation of velocity averaging techniques from their usual L1 context to the natural L2 setting of the stochastic case. We introduce a new mixed kinetic/mild representation of solutions to quasilinear SPDE and use L2 based a priori bounds to treat the stochastic term.

Funding Statement

SB is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), under the project EP/L015684/1. BG acknowledges support by the Max Planck Society through the Max Planck Research Group Stochastic partial differential equations. This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - SFB 1283/2 2021 - 317210226. HW is supported by the Royal Society through the University Research Fellowship UF140187 and by the Leverhulme Trust through a Philip Leverhulme Prize.


SB, BG and HW thank the Isaac Newton Institute for Mathematical Sciences for hospitality during the programme Scaling limits, rough paths, quantum field theory which was supported by EPSRC Grant No. EP/R014604/1.

BG is also affiliated with Max Planck Institute for Mathematics in the Sciences, Leipzig.


Download Citation

Stefano Bruno. Benjamin Gess. Hendrik Weber. "Optimal regularity in time and space for stochastic porous medium equations." Ann. Probab. 50 (6) 2288 - 2343, November 2022.


Received: 1 October 2021; Revised: 1 March 2022; Published: November 2022
First available in Project Euclid: 23 October 2022

MathSciNet: MR4499278
zbMATH: 1500.35072
Digital Object Identifier: 10.1214/22-AOP1583

Primary: 35R60 , 60H15

Keywords: kinetic formulation , Kinetic solution , Stochastic porous medium equations , velocity averaging lemmata

Rights: This research was funded, in whole or in part, by [EPSRC, EP/L015684/1]. A CC BY 4.0 license is applied to this article arising from this submission, in accordance with the grant's open access conditions


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Vol.50 • No. 6 • November 2022
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