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 タイトル: Asymptotic Self-Similarity and Short Time Asymptotics of Stochastic Flows 著者: Kunita, Hiroshi 発行日: 1997年 出版者: Graduate School of Mathematical Sciences, The University of Tokyo 掲載誌情報: Journal of Mathematical Sciences, The University of Tokyo. Vol. 4 (1997), No. 3, Page 595-619 抄録: We study asymptotic properties of Levy flows, changing scales of the space and the time. Let $ξ_t(x), t\geq 0$ be a Levy flow on a Euclidean space ${\bf R}^d$ determined by a SDE driven by an operator stable Levy process. Consider the Levy flows $ξ^{(r)}_t(x)=γ^{(x)}_{1/r}(ξ_{rt}(x)), t\geq 0$, where $\{γ^{(x)}_r\}_{r>0}$ is a dilation, i.e., a one parameter group of diffeomorphisms of ${\bf R}^d$ with invariant point $x$ such that $γ^{(x)}_{1/r}(y)\to \infty$ as $r \to 0$ whenever $y\ne x$. We show that as $r \to 0$ $\{ξ^{(r)}_t(x), t\geq 0\}$ converge weakly to a stochastic flow $\{ξ^{(0)}_t(x), t \geq 0\}$, if we choose a suitable dilation. Further, the limit flow is self-similar with respect to the dilation, i.e., its law is invariant by the above changes of the space and the time. This fact enables us to prove that the short time asymptotics of the density function of the distribution of $ξ_t(x)$ coincides with that of the density function of the distribution of $ξ^{(0)}_t(x)$. URI: http://hdl.handle.net/2261/1361 ISSN: 13405705 出現カテゴリ: Journal of Mathematical Sciences, the University of TokyoJournal of Mathematical Sciences, the University of Tokyo

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