Stochastic flows for Lévy processes with Hölder drifts

In this paper, we study the following stochastic differential equation (SDE) in R^d: {equation presented}, 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 ∈ R^d. Moreover, these strong solutions form a C¹-stochastic flow. As a consequence, we show that, when Z is an α-stable-type Lévy process with α ∈ (0, 2) and b is a bounded β-Hölder continuous function with β ∈ (1 - α/2, 1), the SDE above has a unique strong solution. When α ∈ (0, 1), this in particular partially solves an open problem from Priola. Moreover, we obtain a Bismut type derivative formula for {equation presented} 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: {equation presented}, where L is the generator of the Lévy process Z.