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 $W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right)$.