Stochastic flows for Lévy processes with Hölder drifts

Zhen Qing Chen, Renming Song, Xicheng Zhang

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper, we study the following stochastic differential equation (SDE) in Rd: {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 ∈ Rd. 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.

Original languageEnglish (US)
Pages (from-to)1755-1788
Number of pages34
JournalJournal of Physical Activity and Health
Volume15
Issue number12
DOIs
StatePublished - 2018

Keywords

  • Bismut formula
  • C-diffeomorphism
  • Gradient estimate
  • Pathwise uniqueness
  • SDE
  • Stable process
  • Stochastic flow
  • Strong existence
  • Subcritical
  • Subordinate Brownian motion
  • Supercritical

ASJC Scopus subject areas

  • Orthopedics and Sports Medicine

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