On the $L_p$-boundedness of the stochastic singular integral operators and its application to $L_p$-regularity theory of stochastic partial differential equations

Abstract:

In this article we introduce a stochastic counterpart of the Hörmander condition and Calderón-Zygmund theorem. Let $W_t$ be a Wiener process in a probability space $\Omega$ and let $K(\omega ,r,t,x,y)$ be a random kernel which is allowed to be stochastically singular in a domain $\mathcal {O} \subset \mathbf {R}^d$ in the sense that \begin{equation*} \mathbb {E} \left |\int _0^{t} \int _{|x-y|<\varepsilon }|K(\omega , s, t,y,x)|dy dW_s\right |^p = \infty \quad \forall \, t, p,\varepsilon >0,\, x\in \mathcal {O}. \end{equation*} We prove that the stochastic integral operator of the type \begin{align} \mathbb {T} g(t,x) \coloneq \int _0^{t} \int _{\mathcal {O}} K(\omega ,s,t,y,x) g(s,y)dy dW_s \end{align} is bounded on $\mathbb {L}_p=L_p \left (\Omega \times (0,\infty ); L_{p}(\mathcal {O}) \right )$ for all $p \in [2,\infty )$ if it is bounded on $\mathbb {L}_2$ and the following (which we call stochastic Hörmander condition) holds: there exists a quasi-metric $\rho$ on $(0,\infty )\times \mathcal {O}$ and a positive constant $C_0$ such that for $X=(t,x), Y=(s,y), Z=(r,z) \in (0,\infty ) \times \mathcal {O}$, \begin{equation*} \sup _{\omega \in \Omega ,X,Y}\int _{0}^\infty \left [ \int _{\rho (X,Z) \geq C_0 \rho (X,Y)} | K(r,t, z,x) - K(r,s, z,y)| ~dz\right ]^2 dr <\infty . \end{equation*} Such a stochastic singular integral naturally appears when one proves the maximal regularity of solutions to stochastic partial differential equations (SPDEs). As applications, we obtain the sharp $L_p$-regularity result for a wide class of SPDEs, which includes SPDEs with time measurable pseudo-differential operators and SPDEs defined on non-smooth angular domains.