English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Let $A = -{\rm div} \,a(\cdot) \nabla$ be a second order divergence formelliptic operator on $\R^n$ with bounded measurable real-valued coefficientsand let $W$ be a cylindrical Brownian motion in a Hilbert space $H$. Our mainresult implies that the stochastic convolution process $$ u(t) = \int_0^te^{-(t-s)A}g(s)\,dW(s), \quad t\ge 0,$$ satisfies, for all $1\le p<\infty$, aconical 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 $$\E \n \nabla u \n_{L^p(\R_+;L^2(\R^n;\R^n))}^p \leC^p \E \n g \n_{L^p(\R_+;L^2(\R^n;H))}^p$$ 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 obtainconical stochastic maximal $L^p$-regularity for a class of nonlinear SPDEs withrough initial data.

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