Asymptotic Self-Similarity and Short Time Asymptotics of Stochastic Flows

J. Math. Sci. Univ. Tokyo
Vol. 4 (1997), No. 3, Page 595--619.

Kunita, Hiroshi
Asymptotic Self-Similarity and Short Time Asymptotics of Stochastic Flows
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Abstract:
We study asymptotic properties of Lévy flows, changing scales of the space and the time. Let $ξ_t(x), t\geq 0$ be a Lévy flow on a Euclidean space ${\bf R}^d$ determined by a SDE driven by an operator stable Lévy process. Consider the Lévy 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 $ξ^{(0)}_t(x)$.

Mathematics Subject Classification (1991): 60B15, 60H10, 60J30
Mathematical Reviews Number: MR1484603

Received: 1996-06-11