Conical stochastic maximal $L^p$-regularity for $1 \leq p \lt \infty$

Abstract

Let $A = -{\rm div} \,a(\cdot) \nabla$ be a second order divergence form elliptic operator on $\R^n$ 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 $$ u(t) = \int_0^t e^{-(t-s)A}g(s)\,dW(s), \quad t\ge 0,$$ satisfies, for all $1\le p<\infty$, a conical maximal $L^p$-regularity estimate $$\E \n \nabla u \n_{ T_2^{p,2}(\R_+\times\R^n)}^p \le C_p^p \E \n g \n_{ T_2^{p,2}(\R_+\times\R^n;H)}^p.$$ Here, $T_2^{p,2}(\R_+\times\R^n)$ and $T_2^{p,2}(\R_+\times\R^n;H)$ are the parabolic tent spaces of real-valued and $H$-valued functions, respectively. This contrasts with Krylov's maximal $L^p$-regularity estimate $$\E \n \nabla u \n_{L^p(\R_+;L^2(\R^n;\R^n))}^p \le C^p \E \n g \n_{L^p(\R_+;L^2(\R^n;H))}^p$$ which is known to hold only for $2\le p<\infty$, even when $A = -\Delta$ and $H = \R$. 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.