Stochastic and deterministic parabolic equations with bounded measurable coefficients in space and time: well-posedness and maximal regularity

Pascal Auscher, Pierre Portal

Published 2023-10-10Version 1

We establish well-posedness and maximal regularity estimates for linear parabolic SPDE in divergence form involving random coefficients that are merely bounded and measurable in the time, space, and probability variables. To reach this level of generality, and avoid any of the smoothness assumptions used in the literature, we introduce a notion of pathwise weak solution and develop a new harmonic analysis toolkit. The latter includes techniques to prove the boundedness of various maximal regularity operators on relevant spaces of square functions, the parabolic tent spaces ${\rm T}^{p}$. Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on ${\rm L}^2( \mathbb{R}_+ \times \mathbb{R}^n)={\rm T}^2$ to ${\rm T}^p$, for all $1-\varepsilon<p\le \infty$. As for the estimates related to the stochastic heat equation, we extrapolate in the range $1<p\le \infty$, hence beating the necessary $p\ge 2$ limitation when using the space ${\rm L}^p(\mathbb{R}^n; {\rm L}^2(\mathbb{R}_+))$.