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

TL;DR

This paper proves well-posedness and maximal regularity for linear parabolic stochastic PDEs with very general bounded measurable coefficients, introducing new solution concepts and harmonic analysis tools to handle minimal regularity assumptions.

Contribution

It introduces a notion of pathwise weak solution and develops harmonic analysis techniques to establish well-posedness and regularity for SPDEs with minimal coefficient regularity.

Findings

Established well-posedness for stochastic parabolic PDEs with measurable coefficients.

Extended Lions maximal regularity theorem from L^2 to T^p spaces.

Developed new harmonic analysis toolkit for parabolic tent spaces.

Abstract

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 $\mathrm{T}^{p}$. Applied to deterministic parabolic PDE in divergence form with real coefficients, our results also give the first extension of Lions maximal regularity theorem on $\mathrm{L}^{2}(\mathbb{R}_{+} \times \mathbb{R}^{n})=\mathrm{T}^2$ to $\mathrm{T}^p$, for all $1-\varepsilon<p\le \infty$.