Uniqueness of stable processes with drift
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- by Zhen-Qing Chen and Longmin Wang
- Proc. Amer. Math. Soc. 144 (2016), 2661-2675
- DOI: https://doi.org/10.1090/proc/12909
- Published electronically: October 21, 2015
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Abstract:
Suppose that $d\geq 1$ and $\alpha \in (1, 2)$. Let $\mathcal {L}^b=-(-\Delta )^{\alpha /2} + b \cdot \nabla$, where $b$ is an $\mathbb {R}^d$-valued measurable function on $\mathbb {R}^d$ belonging to a certain Kato class of the rotationally symmetric $\alpha$-stable process $Y$ on $\mathbb {R}^d$. We show that the martingale problem for $(\mathcal {L}^b, C^\infty _c(\mathbb {R}^d))$ has a unique solution for every starting point $x\in \mathbb {R}^d$. Furthermore, we show that the stochastic differential equation $\mathrm {d} X_t=\mathrm {d} Y_t+b(X_t)\mathrm {d} t$ with $X_0=x$ has a unique weak solution for every $x\in \mathbb {R}^d$.References
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Bibliographic Information
- Zhen-Qing Chen
- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 242576
- ORCID: 0000-0001-7037-4030
- Email: zqchen@uw.edu
- Longmin Wang
- Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
- MR Author ID: 776837
- Email: wanglm@nankai.edu.cn
- Received by editor(s): July 7, 2014
- Received by editor(s) in revised form: May 30, 2015, and July 25, 2015
- Published electronically: October 21, 2015
- Additional Notes: The research of the first author was partially supported by NSF Grant DMS-1206276, and NNSFC Grant 11128101
The research of the second author was partially supported by NSFC Grant 11101222, and the Fundamental Research Funds for the Central Universities. - Communicated by: David Levin
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 144 (2016), 2661-2675
- MSC (2010): Primary 60H10, 47G20; Secondary 60G52
- DOI: https://doi.org/10.1090/proc/12909
- MathSciNet review: 3477084