Stochastic flows for Lévy processes with Hölder drifts

In this paper, we study the following stochastic differential equation (SDE) in ${\mathbb{R}}^d$:

where $Z$ is a Lévy process. We show that for a large class of Lévy processes $Z$ and Hölder continuous drifts $b$, the SDE above has a unique strong solution for every starting point $x\in {\mathbb{R}}^d$. Moreover, these strong solutions form a $C^1$-stochastic flow. As a consequence, we show that, when $Z$ is an $\alpha$-stable-type Lévy process with $\alpha \in (0,2)$ and $b$ is a bounded $\beta$-Hölder continuous function with $\beta \in (1-\alpha /2,1)$, the SDE above has a unique strong solution. When $\alpha \in (0,1)$, this in particular partially solves an open problem from Priola. Moreover, we obtain a Bismut type derivative formula for $\nabla {\mathbb{E}}_{x}f(X_t)$ when $Z$ is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with Hölder continuous $b$ and $f$:

where $\mathcal{L}$ is the generator of the Lévy process $Z$.