论文标题
$ l_p $ - 最大的规律性估算二阶随机偏微分方程的解决方案的矩
An $L_p$-maximal regularity estimate of moments of solutions to second-order stochastic partial differential equations
论文作者
论文摘要
我们获得了以下二阶随机偏微分方程（SPDE）的唯一性和解决方案$ u $：\ 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 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 $$ κ|ξ|^2 \ leq \ bar a^{ij}（ω，t）ξ^iξ^iξ^j \ leq k | eq^|^2 \ qquad \ qquad \ forall（ω，t，t，ξ）\ inω\ times（0，t） （0，\ infty）$，$$ f \ in l_p \ left（（（0，t）\ times {\ mathbf {r}}}^d，dt \ times dx; l_r（ω，{\ shatscr {f}}，dp），dp），dp），dp）\ right） {\ Mathbf {r}}^d，dt \ times dx;
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.
