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

TL;DR

This paper establishes weak uniqueness and analyzes the distributional properties of solutions to stochastic differential equations driven by rotationally invariant alpha-stable noise, under specific regularity conditions on the drift.

Contribution

It proves weak uniqueness under a balance condition between stability index and drift regularity, and provides explicit representations and bounds for the transition density and its derivatives.

Findings

Weak uniqueness holds if alpha + gamma > 1.

Transition density exists, is continuous, and has an explicit principal and residual part.

Derivative of the transition density w.r.t. time is also explicitly represented.

Abstract

For an SDE driven by a rotationally invariant $\alpha$-stable noise we prove weak uniqueness of the solution under the balance condition $\alpha+\gamma>1$, where $\gamma$ 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.