Abstract
Abstract For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental solutions, and derive their sharp two-sided estimates. The estimates give explicit dependence on a and recover the sharp estimates for Brownian motion with drift as a→0. Each fundamental solution determines a conservative Feller process X. We characterize X as the unique solution of the corresponding martingale problem as well as a Lévy process with singular drift.
| Original language | English |
|---|---|
| Article number | 2761 |
| Pages (from-to) | 2603-2642 |
| Number of pages | 40 |
| Journal | Stochastic Processes and their Applications |
| Volume | 125 |
| Issue number | 7 |
| DOIs | |
| Publication status | Published - Jul 2015 |
| Externally published | Yes |
Keywords
- Feller semigroup
- Heat kernel
- Kato class
- Lévy system
- Perturbation
- Positivity
- Transition density
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Chen, Z. Q., & Hu, E. (2015). Heat kernel estimates for Δ+Δα/2 under gradient perturbation. Stochastic Processes and their Applications, 125(7), 2603-2642. Article 2761. https://doi.org/10.1016/j.spa.2015.02.016