Stochastic flows for Lévy processes with Hólder drifts.

In this paper, we study the following stochastic differential equation (SDE) in R^d: dXt = dZ_t + b(t,X_t) dt, X0 = x, where Z is a L'evy process. We show that for a large class of L'evy processes Z and H¨older continuous drifts b, the SDE above has a unique strong solution for every starting point x ∈ Rd. Moreover, these strong solutions form a C1-stochastic flow. As a consequence, we show that, when Z is an α-stable-type L'evy process with α ∈ (0, 2) and b is a bounded β-H¨older 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 ∇Exf(Xt) when Z is a subordinate Brownian motion. To study the SDE above, we first study the following nonlocal parabolic equation with H¨older continuous b and f: ∂tu + Lu + b · ∇u + f = 0, u(1, ·) = 0, where L is the generator of the L'evy process Z. [ABSTRACT FROM AUTHOR]