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Uniqueness of stable processes with drift


Authors: Zhen-Qing Chen and Longmin Wang
Journal: Proc. Amer. Math. Soc. 144 (2016), 2661-2675
MSC (2010): Primary 60H10, 47G20; Secondary 60G52
DOI: https://doi.org/10.1090/proc/12909
Published electronically: October 21, 2015
MathSciNet review: 3477084
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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 $ \textup {d} X_t=\textup {d} Y_t+b(X_t)\textup {d} t$ with $ X_0=x$ has a unique weak solution for every $ x\in \mathbb{R}^d$.


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Additional Information

Zhen-Qing Chen
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: zqchen@uw.edu

Longmin Wang
Affiliation: School of Mathematical Sciences, Nankai University, Tianjin 300071, People’s Republic of China
Email: wanglm@nankai.edu.cn

DOI: https://doi.org/10.1090/proc/12909
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
Article copyright: © Copyright 2015 American Mathematical Society

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