二阶随机偏微分方程解的矩的$L_p$ -极大正则性估计
An $L_p$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations
摘要：
本文证明了如下二阶随机偏微分方程解$u$的存在唯一性：\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}其中$T \in (0,\infty)$，$w^k$和$(k=1,2,\ldots)$是独立的Wiener过程，$(\bar a^{ij}(ω,t))$是一个（可预测的）非负对称矩阵值随机过程，使得对于某些$κ, K \in (0,\infty)$，$$ f \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(Ω, {\mathscr{F}} ,dP) \right), $$和$$ g, g_x \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(Ω, {\mathscr{F}} ,dP; l_2) \right) $$，当满足$2 \leq r \leq p < \infty$和适当的可测条件时，有$$ κ|ξ|^2 \leq \bar a^{ij}(ω,t) ξ^i ξ^j \leq K |ξ|^2 \qquad \forall (ω,t,ξ) \in Ω\times (0,T) \times {\mathbf{R}}^d $$.
英文摘要：
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 $T \in (0,\infty)$, $w^k$ $(k=1,2,\ldots)$ are independent Wiener processes, $(\bar a^{ij}(ω,t))$ is a (predictable) nonnegative symmetric matrix valued stochastic process such that $$ κ|ξ|^2 \leq \bar a^{ij}(ω,t) ξ^i ξ^j \leq K |ξ|^2 \qquad \forall (ω,t,ξ) \in Ω\times (0,T) \times {\mathbf{R}}^d $$ for some $κ, K \in (0,\infty)$, $$ f \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(Ω, {\mathscr{F}} ,dP) \right), $$ and $$ g, g_x \in L_p\left( (0,T) \times {\mathbf{R}}^d, dt \times dx ; L_r(Ω, {\mathscr{F}} ,dP; l_2) \right) $$ with $2 \leq r \leq p < \infty$ and appropriate measurable conditions.
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