We establish sharp two-sided bounds on the heat kernel of the fractional Laplacian, perturbed by a drift having critical-order singularity, by transferring it to appropriate weighted space with singular/vanishing weight. We will also talk about weak well-posedness of the corresponding SDE with drift satisfying some minimal assumptions (i.e. such that the corresponding Kolmogorov operator is well-defined in L^2).

The talk is based on joint papers with K.R.Madou, Yu.A.Semenov and K.Szczypkowski