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

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 R^d: dX_t=b(t,X_t)dt+σ(t,X_t)dW_t,X₀=x, where σ is bounded, uniformly continuous and nondegenerate, b∈L˜_q1^p₁ and ∇σ∈L˜_q2^p₂ for some p_i,q_i∈[2,∞) with [Formula presented], i=1,2, where L˜_qi^p_i,i=1,2 are some localized spaces of L^q_i(R₊;L^p_i(R^d)). Moreover, in the endpoint case b∈L˜_∞^d;uni⊂L˜_∞^d, we also show the weak well-posedness.