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Conservativeness of Diffusion Processes with Drift

Kazuhiro Kuwae
Proceedings of the American Mathematical Society
Vol. 132, No. 9 (Sep., 2004), pp. 2743-2751
Stable URL: http://www.jstor.org/stable/4097393
Page Count: 9
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Conservativeness of Diffusion Processes with Drift
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Abstract

We show the conservativeness of the Girsanov transformed diffusion process by drift $b \in L^p (\mathbb{R}^d \rightarrow \mathbb{R}^d)$ with $p \geq 4/(2 - \sqrt{2\delta(\left\vert b\right\vert^2)/\lambda)}$ or p > 4d/(d+2), or p = 2 if $\left\vert b\right\vert^2$ is of the Hardy class with sufficiently small coefficient of energy $\delta(\left\vert b\right\vert^2) < \lambda/2$. 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 $b \in L^d (\mathbb{R}^d \rightarrow \mathbb{R}^d$ when d ≥ 3.

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