# Revista Matemática Iberoamericana

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**Volume 34, Issue 4, 2018, pp. 1755–1788**

**DOI: 10.4171/rmi/1042**

Published online: 2018-12-03

Stochastic flows for Lévy processes with Hölder drifts

Zhen-Qing Chen^{[1]}, Renming Song

^{[2]}and Xicheng Zhang

^{[3]}(1) University of Washington, Seattle, USA

(2) University of Illinois at Urbana-Champaign, USA

(3) Wuhan University, Wuhan, Hubei, China

In this paper, we study the following stochastic differential equation (SDE) in $\mathbb R^d$: $$\mathrm d X_t= \mathrm d Z_t + b(t, X_t)\,\mathrm d t, \quad X_0=x,$$ 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 $f$: $$\partial_t u+\mathscr L u+b\cdot \nabla u+f=0,\quad u(1, \cdot )=0,$$ where $\mathscr L$ is the generator of the Lévy process $Z$.

*Keywords: *SDE, supercritical, subcritical, stable process, subordinate Brownian motion, gradient estimate, strong existence, pathwise uniqueness, Bismut formula, stochastic flow, $C^1$-diffeomorphism.

Chen Zhen-Qing, Song Renming, Zhang Xicheng: Stochastic flows for Lévy processes with Hölder drifts. *Rev. Mat. Iberoam.* 34 (2018), 1755-1788. doi: 10.4171/rmi/1042