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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Conservativeness of diffusion processes with drift

Author: Kazuhiro Kuwae
Journal: Proc. Amer. Math. Soc. 132 (2004), 2743-2751
MSC (2000): Primary 60J45; Secondary 31C25
Published electronically: April 21, 2004
MathSciNet review: 2054801
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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 $p\geq4/(2-\sqrt{2\delta(\vert b\vert^2)/\lambda})$ 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 $d\geq3$.

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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

Keywords: Semi-Dirichlet form, Dirichlet form, diffusion process, Kato class function, Hardy class function, Sobolev inequality, Novikov's condition, supermartingale, exponential martingale, conservativeness, Girsanov transformation
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
Article copyright: © Copyright 2004 American Mathematical Society