On weak uniqueness and distributional properties of a solution to an SDE with $α$-stable noise

Abstract

For an SDE driven by a rotationally invariant $α$-stable noise we prove weak uniqueness of the solution under the balance condition $α+γ>1$, where $γ$ denotes the Holder index of the drift coefficient. We prove existence and continuity of the transition probability density of the corresponding Markov process and give a representation of this density with an explicitly given "principal part", and a "residual part" which possesses an upper bound. Similar representation is also provided for the derivative of the transition probability density w.r.t. the time variable.