Conservativeness of diffusion processes with drift
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Abstract:
We show the conservativeness of the Girsanov transformed diffusion process by drift $b\in L^p(\mathbb {R}^d\!\to \!\mathbb {R}^d)$ with $p\geq 4/(2-\sqrt {2\delta (\vert b\vert ^2)/\lambda })$ or $p>4d/(d+2)$, or $p=2$ if $\vert b\vert ^2$ is of the Hardy class with sufficiently small coefficient of energy $\delta (\vert b\vert ^2)<\lambda /2$. Here $\lambda >0$ is the lower bound of the symmetric measurable matrix-valued function $a(x):=(a_{i,j}(x))_{i,j}$ appearing in the given Dirichlet form. In particular, our result improves the conservativeness of the transformed process by $b\in L^d(\mathbb {R}^d\!\to \!\mathbb {R}^d)$ when $d\geq 3$.References
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Additional Information
- Kazuhiro Kuwae
- Affiliation: Department of Mathematical Sciences, Yokohama City University, Yokohama 236-0027, Japan
- Address at time of publication: Department of Mathematics, Faculty of Education, Kumamoto University, Kumamoto 860-8555, Japan
- Email: kuwae@yokohama-cu.ac.jp, kuwae@gpo.kumamoto-u.ac.jp
- Received by editor(s): June 18, 2002
- Received by editor(s) in revised form: December 20, 2002
- Published electronically: April 21, 2004
- Additional Notes: The author was partially supported by a Grant-in-Aid for Scientific Research (C) No. 13640220 from the Japanese Ministry of Education, Culture, Sports, Science and Technology
- Communicated by: Richard C. Bradley
- © Copyright 2004 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 132 (2004), 2743-2751
- MSC (2000): Primary 60J45; Secondary 31C25
- DOI: https://doi.org/10.1090/S0002-9939-04-07283-1
- MathSciNet review: 2054801