Drift perturbation of subordinate Brownian motions with Gaussian component

Zhen Qing Chen, Xiao Man Dou*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)

Abstract

Let d ≥ 1 and Z be a subordinate Brownian motion on Rd with infinitesimal generator Δ + ψ(Δ), where ψ is the Laplace exponent of a one-dimensional non-decreasing Lévy process (called subordinator). We establish the existence and uniqueness of fundamental solution (also called heat kernel) pb(t, x, y) for non-local operator ℒb = Δ + ψ(Δ) + b · ∇, where b is an Rd-valued function in Kato class Kd,1. We show that pb(t, x, y) is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for (Lb,Cc (Rd)) and X is a weak solution of (Formula presented. ). Moreover, we prove that the above stochastic differential equation has a unique weak solution.

Original languageEnglish
Pages (from-to)239-260
Number of pages22
JournalScience China Mathematics
Volume59
Issue number2
DOIs
Publication statusPublished - 1 Feb 2016
Externally publishedYes

Keywords

  • Feller process
  • Kato class
  • Lévy system
  • gradient perturbation
  • heat kernel
  • martingale problem
  • stochastic differential equation
  • subordinate Brownian motion

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