Stochastic flows for Lévy processes with Hölder drifts

• Zhen-Qing Chen ^[1] ; Renming Song ^[2] ; Xicheng Zhang ^[3]
  1. [1] University of Washington

     University of Washington

     Estados Unidos

     
  2. [2] University of Illinois at Urbana Champaign

     University of Illinois at Urbana Champaign

     Township of Cunningham, Estados Unidos

     
  3. [3] Wuhan University

     Wuhan University

     China

     

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• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 34, Nº 4, 2018, págs. 1755-1788
• Idioma: inglés
• DOI: 10.4171/rmi/1042
• Texto completo no disponible (Saber más ...)
• Resumen

  • In this paper, we study the following stochastic differential equation (SDE) in R d Rd: dX t =dZ t +b(t,X t )dt,X 0 =x, whereZ 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 ∇E x f(X t ) ∇Exf(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:

    ∂ t u+Lu+b⋅∇u+f=0,u(1,⋅)=0