By establishing a characterization for Sobolev differentiability of random fields, we prove the weak differentiability of solutions to stochastic differential equations with local Sobolev and super-linear growth coefficients with respect to the starting point. Moreover, we also study the strong Feller property and the irreducibility to the associated diffusion semigroup.
"Sobolev differentiable flows of SDEs with local Sobolev and super-linear growth coefficients." Ann. Probab. 44 (6) 3661 - 3687, November 2016. https://doi.org/10.1214/15-AOP1057