[Une note sur un seuil pour la régularité en temps d’EDP stochastiques]
We consider solutions to linear parabolic SPDEs of the form
| \[ \mathrm{d} u(t) + A u(t) \, \mathrm{d} t = g(t) \, \mathrm{d} \beta , \qquad u(0)=0, \] |
where $A$ is a positive, invertible, and self-adjoint operator on a Hilbert space $X$, $\beta $ is a one-dimensional Brownian motion, and $g(t)\equiv x\in X$. We show that, for all $\alpha \in \bigl [0,\frac{1}{2}\bigr )$,
| \[ u\in L^2\bigl (\Omega ;W^{\alpha ,2}\bigl (0,T;\operatorname{D}(A^{1/2})\bigr )\bigr ) \quad \text{if and only if} \quad x\in \operatorname{D}(A^{\alpha }). \] |
In particular, there is a lack of persistence of temporal regularity from the diffusion coefficient $g$ to the solution, and additional spatial regularity is required to improve time regularity. In particular, this provides a counterexample to a conjectured time-regularity property for monotone stochastic evolution equations posed by D. Breit and M. Hofmanová in [Comptes Rendus. Mathématique 354 (2016)].
Nous considérons les solutions d’équations aux dérivées partielles stochastiques (EDPS) paraboliques linéaires de la forme
| \[ \mathrm{d} u(t) + A u(t) \, \mathrm{d} t = g(t) \, \mathrm{d} \beta , \qquad u(0)=0, \] |
où $A$ est un opérateur positif, inversible et auto-adjoint sur un espace de Hilbert $X$, $\beta $ est un mouvement brownien unidimensionnel, et $g(t)\equiv x\in X$. Nous montrons que, pour tout $\alpha \in \bigl [0,\frac{1}{2}\bigr )$,
| \[ u\in L^2\bigl (\Omega ;W^{\alpha ,2}\bigl (0,T;\operatorname{D}(A^{1/2})\bigr )\bigr ) \quad \text{si et seulement si} \quad x\in \operatorname{D}(A^{\alpha }). \] |
En particulier, il n’y a pas de persistance de la régularité temporelle du coefficient de diffusion $g$ vers la solution, et une régularité spatiale supplémentaire est nécessaire pour améliorer la régularité temporelle. En particulier, cela fournit un contre-exemple à une propriété de régularité temporelle conjecturée pour les équations d’évolution stochastiques monotones formulée par D. Breit et M. Hofmanová dans [Comptes Rendus. Mathématique 354 (2016)].
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Keywords: Time regularity, parabolic stochastic PDEs, fractional smoothness
Mots-clés : Régularité temporelle, équations différentielles partielles stochastiques paraboliques, régularité fractionnaire
Antonio Agresti 1 ; Mark C. Veraar 2
CC-BY 4.0
Antonio Agresti; Mark C. Veraar. A note on a threshold for temporal regularity of stochastic PDEs. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 371-379. doi: 10.5802/crmath.833
@article{CRMATH_2026__364_G2_371_0,
author = {Antonio Agresti and Mark C. Veraar},
title = {A note on a threshold for temporal regularity of stochastic {PDEs}},
journal = {Comptes Rendus. Math\'ematique},
pages = {371--379},
year = {2026},
publisher = {Acad\'emie des sciences, Paris},
volume = {364},
doi = {10.5802/crmath.833},
language = {en},
}
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