Stochastic maximal \(L^{p}\)-regularity (Q414290)

Properties of stochastic convolutions \(U(t)=\int_0^tS(t-s)G(s)\,dW_H(s)\), \(t\geq 0\), in \(L^q(\mathcal O)\)-spaces over \(\sigma\)-finite measure spaces are studied. Here \(q\in[2,\infty)\), \(S\) is a bounded analytic semigroup on \(L^q(\mathcal O)\) generated by an infinitesimal generator \(-A\) such that \(A\) admits a bounded \(H^\infty\)-calculus of angle less than \(\pi/2\) on \(L^q(\mathcal O)\), \(G\) is a measurable adapted process in \(L^p(\mathbb R_+\times\Omega;L^q(\mathcal O;H))\), \(p\in(2,\infty)\), and \(W_H\) is a cylindrical Wiener process on a real separable Hilbert space \(H\). It is proved that the stochastic convolution \(U\) is a well-defined process in \(L^q(\mathcal O)\), takes values in \(\text{Dom}(A^{1/2})\) almost surely and that a maximal \(L^p\)-estimate NEWLINE\[NEWLINE\operatorname{E}\|A^{1/2}U\|^p_{L^p(\mathbb R_+;L^q(\mathcal O))}\leq C^p\operatorname{E}\|G\|^p_{L^p(\mathbb R_+;L^q(\mathcal O;H))}NEWLINE\]NEWLINE holds with a constant \(C\) independent of \(G\) and \(p\in(2,\infty)\). This estimate also holds for \(p=q=2\), whereas counterexamples to its validity for \((p,q)\in\{2\}\times(2,\infty)\) are provided. If, in addition, \(A\) is invertible on \(L^q(\mathcal O)\) and \(\theta\in[0,1/2)\), then NEWLINE\[NEWLINE\operatorname{E}[\|U\|^p_{H^{\theta,p}(\mathbb R_+;D(A^{1/2-\theta}))}+\sup_{t\geq 0}\|U\|^p_{D_A(1/2-1/p)}]\leq C^p\operatorname{E}\|G\|^p_{L^p(\mathbb R_+;L^q(\mathcal O;H))}NEWLINE\]NEWLINE is proved to hold for some constant \(C\) independent of \(G\).NEWLINENEWLINEThe assumptions on \(A\) are satisfied by elliptic differential operators appearing usually in partial differential equations (examples of applicable operators \(A\) are surveyed in the paper) and the proofs take advantage of deterministic tools such as McIntosh's \(H^\infty\)-calculus, \(R\)-boundedness techniques and a theory of stochastic integration in UMD Banach spaces developed in the last decade by the three authors.