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

Abstract: 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 $\mathbb R^d$: $$ {\rm d} X_t=b(t,X_t){\rm d} t+\sigma(t,X_t){\rm d} W_t,\ \ X_0=x\in\mathbb R^d, $$ where $\sigma$ is bounded, uniformly continuous and nondegenerate, $\nabla\sigma\in \widetilde{\mathbb L}^{p_1}_{q_1}$ 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.