On admissible singular drifts of symmetric $\alpha$-stable process

Abstract

We consider the problem of existence of a (unique) weak solution to the SDE describing symmetric $\alpha$-stable process with a locally unbounded drift $b:\mathbb R^d \rightarrow \mathbb R^d$, $d \geq 3$, $1<\alpha<2$. In this paper, $b$ belongs to the class of weakly form-bounded vector fields. The latter arises as the class providing the $L^2$ theory of the non-local operator behind the SDE, i.e.\,$(-\Delta)^{\frac{\alpha}{2}} + b \cdot \nabla$, and contains as proper sub-classes the other classes of singular vector fields studied in the literature in connection with this operator, such as the Kato class, weak $L^{\frac{d}{\alpha-1}}$ class and the Campanato-Morrey class (thus, $b$ can be so singular that it destroys the standard heat kernel estimates in terms of the heat kernel of the fractional Laplacian). We show that for such $b$ the operator $-(-\Delta)^{\frac{\alpha}{2}} - b \cdot \nabla$ admits a realization as a Feller generator, and that the probability measures determined by the Feller semigroup (uniquely in appropriate sense) admit description as weak solutions to the corresponding SDE. The proof is based on detailed regularity theory of $(-\Delta)^{\frac{\alpha}{2}} + b \cdot \nabla$ in $L^p$, $p>d-\alpha+1$.