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: $$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t, \quad t \in (0,T); \quad u(0,\cdot )=0, \end{aligned}$$ (1) 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 $$\begin{aligned} \kappa |\xi |^2 \le {\bar{a}}^{ij}(\omega ,t) \xi ^i \xi ^j \le K |\xi |^2 \quad \forall \;(\omega ,t,\xi ) \in \Omega \times (0,T) \times {\mathbf {R}}^d \end{aligned}$$for some \(\kappa , K \in (0,\infty )\), $$\begin{aligned} f \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP) \right) , \end{aligned}$$and $$\begin{aligned} g, g_x \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP; l_2) \right) \end{aligned}$$with \(2 \le r \le p < \infty \) and appropriate measurable conditions. Moreover, for the solution u, we obtain the following maximal regularity moment estimate $$\begin{aligned}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u_{xx}(t,x)|^r\right] \right) ^{p/r} dx dt \nonumber \\&\le N \bigg (\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |f(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \nonumber \\&\quad + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g_x(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \bigg ), \end{aligned}$$ (2) where N is a positive constant depending only on d, p, r, \(\kappa \), K, and T. As an application, for the solution u to (1), the rth moment \(m^r(t,x):=\mathbb {E}|u(t,x)|^r\) is in the parabolic Sobolev space \(W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) \).