Stochastic flows for Lévy processes with Hölder drifts

In this paper, we study the following stochastic differential equation (SDE) in R: dXt = dZt + b(t, Xt)dt, X0 = x, where Z is a Lévy process. We show that for a large class of Lévy processes Z and Hölder continuous drift b, the SDE above has a unique strong solution for every starting point x ∈ R . Moreover, these strong solutions form a C-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 [18]. Moreover, we obtain a Bismut type derivative formula for ∇Ex f (Xt) 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 : ∂tu + L u + b · ∇u + f = 0, u(1, ·) = 0, where L is the generator of the Lévy process Z.