Asymptotic Self-Similarity and Short Time Asymptotics of Stochastic Flows

概要

論文の詳細を見る
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 $ξ^{(0)}_t(x)$.
東京大学の論文