Stochastic flows for L\'evy processes with H\"{o}lder drifts

Abstract

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 L\'evy process. We show that for a large class of L\'evy processes ${Z}$ and H\"older continuous drift $b$, 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 L\'evy process with $\alpha\in (0, 2)$ and $b$ is bounded and $\beta$-H\"older continuous with $\beta\in (1- {\alpha}/{2},1)$, the SDE above has a unique strong solution. When $\alpha \in (0, 1)$, this in particular solves an open problem from Priola \cite{Pr1}. 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 H\"older 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 L\'evy process ${Z}$.