On some applications of Sobolev flows of SDEs with unbounded drift coefficients

We study two applications of spatial Sobolev smoothness of stochastic flows of unique strong solution to stochastic differential equations (SDEs) with irregular drift coefficients. First, we analyse the stochastic transport equation assuming that the drift coefficient is Borel measurable, with spatial linear growth and show that the above equation has a unique Sobolev differentiable weak coefficient for all t∈[0,T] for T small enough. Second, we consider the Kolmogorov equation and obtain a representation of the spatial derivative of its solution v. The latter result is obtained via the martingale representation theorem given in (Elliott and Kohlmann, 1988) and generalises the results in (Elworthy and Li, 1994; Menoukeu-Pamen et al., 2013).