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We establish well-posedness and maximal regularity estimates for linearparabolic SPDE in divergence form involving random coefficients that are merelybounded and measurable in the time, space, and probability variables. To reachthis level of generality, and avoid any of the smoothness assumptions used inthe literature, we introduce a notion of pathwise weak solution and develop anew harmonic analysis toolkit. The latter includes techniques to prove theboundedness of various maximal regularity operators on relevant spaces ofsquare functions, the parabolic tent spaces $\mathrm{T}^{p}$. Applied todeterministic parabolic PDE in divergence form with real coefficients, ourresults 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 $1-\varepsilon<p\le \infty$.

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