An $L_p$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations

We obtain uniqueness and existence of a solution $u$ to the following second-order stochastic partial differential equation (SPDE) : \begin{align} \label{abs eqn} du= \left( \bar a^{ij}(\omega,t)u_{x^ix^j}+ f \right)dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot)=0, \end{align} where $T \in (0,\infty)$, $w^k$ $(k=1,2,\ldots)$ are independent Wiener processes, $(\bar a^{ij}(\omega,t))$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\kappa |\xi|^2 \leq \bar a^{ij}(\omega,t) \xi^i \xi^j \leq K |\xi|^2 \qquad \forall (\omega,t,\xi) \in \Omega \times (0,T) \times {\mathbf{R}}^d$$ for some $\kappa, K \in (0,\infty)$, $$f \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(\Omega, {\mathscr{F}} ,dP) \right),$$ and $$g, g_x \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(\Omega, {\mathscr{F}} ,dP; l_2) \right)$$ with $2 \leq r \leq p < \infty$ and appropriate measurable conditions.