## 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
We study asymptotic properties of LÃ©vy flows, changing scales of the space and the time. Let $Î¾_t(x), t\geq 0$ be a LÃ©vy flow on a Euclidean space ${\bf R}^d$ determined by a SDE driven by an operator stable LÃ©vy process. Consider the LÃ©vy flows $Î¾^{(r)}_t(x)=Î³^{(x)}_{1/r}(Î¾_{rt}(x)), t\geq 0$, where $\{Î³^{(x)}_r\}_{r>0}$ is a dilation, i.e., a one parameter group of diffeomorphisms of ${\bf R}^d$ with invariant point $x$ such that $Î³^{(x)}_{1/r}(y)\to \infty$ as $r \to 0$ whenever $y\ne x$. We show that as $r \to 0$ $\{Î¾^{(r)}_t(x), t\geq 0\}$ converge weakly to a stochastic flow $\{Î¾^{(0)}_t(x), t \geq 0\}$, 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 $Î¾_t(x)$ coincides with that of the density function of the distribution of $Î¾^{(0)}_t(x)$.