Optimal regularity in time and space for stochastic porous medium equations

Stefano Bruno, Benjamin Gess, Hendrik Weber

Research output: Contribution to journalArticlepeer-review

Abstract

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.

Original languageEnglish
Pages (from-to)2288-2343
Number of pages56
JournalAnnals of Probability
Volume50
Issue number6
Early online date23 Oct 2022
DOIs
Publication statusPublished - 30 Nov 2022

Bibliographical note

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.

Funding Information:
Acknowledgments. 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.

Funding Information:
2022, Vol. 50, No. 6, 2288–2343 https://doi.org/10.1214/22-AOP1583 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

Funding Information:
Funding. 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 Forschungs-gemeinschaft (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.

Keywords

  • Kinetic formulation
  • Kinetic solution
  • Stochastic porous medium equations
  • Velocity averaging lemmata.

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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