Stochastics and Partial Differential Equations: Analysis and Computations ( IF 1.4 ) Pub Date : 2021-06-12 , DOI: 10.1007/s40072-021-00201-1 Ildoo Kim
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 \(W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) \).
中文翻译：
$$L_p$$ L p -二阶随机偏微分方程解矩的最大正则性估计
我们得到以下二阶随机偏微分方程解u的唯一性和存在性：
$$\begin{aligned} du= \left( {\bar{a}}^{ij}(\omega ,t)u_{x^ix^j}+ f \right) dt + g^k dw^k_t , \quad t \in (0,T); \quad u(0,\cdot )=0, \end{对齐}$$ (1)其中\(T \in (0,\infty )\) , \(w^k\) \((k=1,2,\ldots )\)是独立的维纳过程，\(({\bar{a} }^{ij}(\omega ,t))\)是一个（可预测的）非负对称矩阵值随机过程，使得
$$\begin{对齐} \kappa |\xi |^2 \le {\bar{a}}^{ij}(\omega ,t) \xi ^i \xi ^j \le K |\xi |^ 2 \quad \forall \;(\omega ,t,\xi ) \in \Omega \times (0,T) \times {\mathbf {R}}^d \end{aligned}$$对于某些\(\kappa , K \in (0,\infty )\)，
$$\begin{aligned} f \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP ) \right) , \end{对齐}$$和
$$\begin{aligned} g, g_x \in L_p\left( (0,T) \times {\mathbf {R}}^d, dt \times dx ; L_r(\Omega , {\mathscr {F}} ,dP; l_2) \right) \end{对齐}$$与\(2 \le r \le p < \infty \)和适当的可测量条件。此外，对于解u，我们获得以下最大正则矩估计
$$\begin{对齐}&\int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u(t,x)|^r\right ] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |u_{xx}(t ,x)|^r\right] \right) ^{p/r} dx dt \nonumber \\&\le N \bigg (\int _0^T \int _{{\mathbf {R}}^d} \left( \mathbb {E}\left[ |f(t,x)|^r\right] \right) ^{p/r} dx dt + \int _0^T \int _{{\mathbf {R }}^d}\left( \mathbb {E}\left[ |g(t,x)|_{l_2}^r\right] \right) ^{p/r} dx dt \nonumber \\&\ quad + \int _0^T \int _{{\mathbf {R}}^d}\left( \mathbb {E}\left[ |g_x(t,x)|_{l_2}^r\right] \右) ^{p/r} dx dt \bigg ), \end{aligned}$$ (2)其中N是一个正常数，仅取决于d、p、r、\(\kappa \)、K和T。作为应用，对于(1)的解u，第r个矩\(m^r(t,x):=\mathbb {E}|u(t,x)|^r\)在抛物线索博列夫空间\(W_{p/r}^{1,2}\left( (0,T) \times \mathbf {R}^d\right) \)。
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