On time regularity of stochastic evolution equations with monotone coefficients

Dominic Breit; Martina Hofmanová

doi:10.1016/j.crma.2015.09.031

Partial differential equations/Probability theory

On time regularity of stochastic evolution equations with monotone coefficients

Presented by: the Editorial Board

Dominic Breit ¹; Martina Hofmanová ²

¹ Department of Mathematics, Heriot-Watt University, Riccarton Edinburgh EH14 4AS, UK
² Technical University Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany

Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 33-37.

Abstract
Résumé

We report on a time regularity result for stochastic evolutionary PDEs with monotone coefficients. If the diffusion coefficient is bounded in time without additional space regularity, we obtain a fractional Sobolev-type time regularity of order up to $\frac{1}{2}$ for a certain functional $G (u)$ of the solution. Namely, $G (u) = \nabla u$ in the case of the heat equation and $G (u) = {| \nabla u |}^{\frac{p - 2}{2}} \nabla u$ for the p-Laplacian. The motivation is twofold. On the one hand, it turns out that this is the natural time regularity result that allows us to establish the optimal rates of convergence for numerical schemes based on a time discretization. On the other hand, in the linear case, i.e. when the solution is given by a stochastic convolution, our result complements the known stochastic maximal space–time regularity results for the borderline case not covered by other methods.

On étudie des résultats de régularité en temps pour des équations aux dérivées partielles stochastiques à coefficients monotones. Si le coefficient de diffusion est borné en temps, sans faire d'hypothèses supplémentaires sur la régularité en espace, on obtient une régularité en temps de type Sobolev fractionnaire d'ordre $\frac{1}{2}$ pour une certaine fonction $G (u)$ de la solution u. Plus précisément, $G (u) = \nabla u$ dans le cas de l'équation de la chaleur et $G (u) = {| \nabla u |}^{\frac{p - 2}{p}} \nabla u$ pour le p-laplacien. La motivation est double : d'une part, il apparaît que ceci correspond à un résultat naturel de régularité en temps et, de plus, on obtient les taux de convergence optimaux pour les schémas de discrétisation en temps ; d'autre part, dans le cas linéaire, c'est-à-dire dans celui où la solution est donnée par une convolution stochastique, le résultat obtenu complète les résultats connus de régularité maximale dans l'espace-temps pour le cas limite, résultats qu'on ne peut pas obtenir par d'autres méthodes.

Received: 2015-08-10
Accepted: 2015-09-30
Published online: 2015-11-03

DOI: 10.1016/j.crma.2015.09.031

Author's affiliations:

Dominic Breit ¹; Martina Hofmanová ²

¹ Department of Mathematics, Heriot-Watt University, Riccarton Edinburgh EH14 4AS, UK
² Technical University Berlin, Institute of Mathematics, Straße des 17. Juni 136, 10623 Berlin, Germany

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     title = {On time regularity of stochastic evolution equations with monotone coefficients},
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     pages = {33--37},
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     year = {2016},
     doi = {10.1016/j.crma.2015.09.031},
     language = {en},
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Dominic Breit; Martina Hofmanová. On time regularity of stochastic evolution equations with monotone coefficients. Comptes Rendus. Mathématique, Volume 354 (2016) no. 1, pp. 33-37. doi : 10.1016/j.crma.2015.09.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.09.031/

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