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arXiv 1908.01255math.PR
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$L^q(L^p)$ -随机微分方程理论

$L^q(L^p)$-theory of stochastic differential equations

Pengcheng Xia, Longjie Xie, Xicheng Zhang, Guohuan Zhao

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摘要：

本文证明了$\mathbb R^d$中下列随机微分方程的唯一强解关于$x$的弱可微性以及Bismut-Elworthy-Li的导数公式：$$ {\rm d} X_t=b(t,X_t){\rm d} t+σ(t,X_t){\rm d} W_t,\ \ X_0=x\in\mathbb R^d, $$，其中$σ$是有界的，一致连续的，非退化的;对于某些$p_i,q_i\in[2,\infty)$，当$\frac{d}{p_i}+\frac{2}{q_i}<1$，$i=1,2$，其中$\widetilde{\mathbb L}^{p_i}_{q_i}, i=1,2$是某些局部化空间时的$\nablaσ\in \widetilde{\mathbb L}^{p_1}_{q_1}$和$b\in \widetilde{\mathbb L}^{p_2}_{q_2}$.此外，在端点情形$b\in \widetilde{\mathbb L}^{d; {\rm uni}}_\infty$，我们也证明了弱适定性。

英文摘要：

In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\mathbb R^d$: $$ {\rm d} X_t=b(t,X_t){\rm d} t+σ(t,X_t){\rm d} W_t,\ \ X_0=x\in\mathbb R^d, $$ where $σ$ is bounded, uniformly continuous and nondegenerate, $\nablaσ\in \widetilde{\mathbb L}^{p_1}_{q_1}$ and $b\in \widetilde{\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\in[2,\infty)$ with $\frac{d}{p_i}+\frac{2}{q_i}<1$, $i=1,2$, where $\widetilde{\mathbb L}^{p_i}_{q_i}, i=1,2$ are some localized spaces. Moreover, in the endpoint case $b\in \widetilde{\mathbb L}^{d; {\rm uni}}_\infty$, we also show the weak well-posedness.

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