Strong regularization by noise for a class of kinetic SDEs driven by symmetric {\alpha}-stable processes

TL;DR

This paper proves strong well-posedness for a class of kinetic SDEs driven by symmetric -stable processes, extending existing results under certain regularity conditions and employing advanced transform techniques.

Contribution

It introduces novel well-posedness results for degenerate kinetic SDEs driven by symmetric -stable processes, including new thresholds for regularity and specialized transforms.

Findings

Established strong well-posedness under Hlder regularity.

Partially recovered optimal thresholds for weak uniqueness.

Developed new transform techniques for different dimensions.

Abstract

We establish strong well-posedness for a class of degenerate SDEs of kinetic type with autonomous diffusion driven by a symmetric $\alpha$-stable process under H\"older regularity conditions for the drift term. We partially recover the thresholds for the H\"older regularity that are optimal for weak uniqueness. In general dimension, we only consider $\alpha = 2$ and need an additional integrability assumption for the gradient of the drift: this condition is satisfied by Peano-type functions. In the one-dimensional case we do not need any additional assumption. In the multi-dimensional case, the proof is based on a first-order Zvonkin transform/PDE, while for the one-dimensional case we use a second-order Zvonkin/PDE transform together with a Watanabe-Yamada technique.