Conical stochastic maximal $L^p$-regularity for $1\leqslant p<\infty$

Abstract

Let $A = -\mathrm{div} \,a(\cdot ) \nabla$ be a second order divergence form elliptic operator on ${\mathbb R}^n$ with bounded measurable real-valued coefficients and let $W$ be a cylindrical Brownian motion in a Hilbert space $H$. Our main result implies that the stochastic convolution process

$$\begin{aligned} u(t) = \int _0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\geqslant 0, \end{aligned}$$

satisfies, for all $1\leqslant p<\infty$, a conical maximal $L^p$-regularity estimate

$$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)}^p \leqslant C_p^p {\mathbb E}\Vert g \Vert _{ T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)}^p. \end{aligned}$$

Here, $T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n)$ and $T_2^{p,2}({\mathbb R}_+\times {\mathbb R}^n;H)$ are the parabolic tent spaces of real-valued and $H$-valued functions, respectively. This contrasts with Krylov’s maximal $L^p$-regularity estimate

$$\begin{aligned} {\mathbb E}\Vert \nabla u \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;{\mathbb R}^n))}^p \leqslant C^p {\mathbb E}\Vert g \Vert _{L^p({\mathbb R}_+;L^2({\mathbb R}^n;H))}^p \end{aligned}$$

which is known to hold only for $2\leqslant p<\infty$, even when $A = -\Delta$ and $H = {\mathbb R}$. The proof is based on an $L^2$-estimate and extrapolation arguments which use the fact that $A$ satisfies suitable off-diagonal bounds. Our results are applied to obtain conical stochastic maximal $L^p$-regularity for a class of nonlinear SPDEs with rough initial data.