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

Abstract

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}(ω,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 $T\in (0,\infty )$, $w^k$ $(k=1,2,\dots )$ are independent Wiener processes, $({\bar{a}}^{ij}(\ddot{I}‰,t))$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$\hat{I}º|\hat{I}¾{|}^2\le {\bar{a}}^{ij}(\ddot{I}‰,t)\hat{I}{¾}^i\hat{I}{¾}^j\le K|\hat{I}¾{|}^2\forall (\ddot{I}‰,t,\hat{I}¾)\in \hat{I}\text{c◯}\times (0,T)\times {\mathbf{R}}^d$$ for some $\hat{I}º,K\in (0,\infty )$, $$f\in L_p\left((0,T)\times {\mathbf{R}}^d,dt\times dx;L_r(\hat{I}\text{c◯},\mathcal{F},dP)\right),$$ and $$g,g_x\in L_p\left((0,T)\times {\mathbf{R}}^d,dt\times dx;L_r(\hat{I}\text{c◯},\mathcal{F},dP;l_2)\right)$$ with $2\le r\le p<\infty$ and appropriate measurable conditions.