Stable random fields and geometry

Volume 90 / 2010

Shigeo Takenaka Banach Center Publications 90 (2010), 225-241 MSC: 60G52, 60G60, 60G51, 53C65. DOI: 10.4064/bc90-0-15

Abstract

Let $(M,d)$ be a metric space with a fixed origin ${\bf O}$. P. Lévy defined Brownian motion $\{X(a);a\in M\}$ as

0. $X({\bf O})=0$.

1. $X(a)-X(b)$ is subject to the Gaussian law of mean $0$ and variance $d(a,b)$.

He gave an example for $M=S^m$, the $m$-dimensional sphere. Let $\{Y(B);B\in {\cal B}(S^m)\}$ be the Gaussian random measure on $S^m$, that is,

1. $\{Y(B)\}$ is a centered Gaussian system,

2. the variance of $Y(B)$ is equal of $\mu(B)$, where $\mu$ is the uniform measure on $S^m$,

3. if $B_1 \cap B_2=\emptyset$ then $Y(B_1)$ is independent of $Y(B_2)$.

4. for $B_i,\ i=1,2,\ldots,$ $B_i \cap B_j =\emptyset, i\ne j$, we have $Y(\cup B_i ) = \sum Y(B_i)$, a.e.

Set $S_a = H_a \triangle H_{\bf O},$ where $H_a$ is the hemisphere with center $a$, and $\triangle$ means symmetric difference. Then $$ \{X(a) = Y(S_a); a\in S^m \} $$ is Lévy's Brownian motion.

In the case of $M=R^m$, $m$-dimensional Euclidean space, N. N. Chentsov showed that $ \{X(a) = Y(S_a)\}$ is an $R^m$-parameter Brownian motion in the sense of P. Lévy. Here $S_a$ is the set of hyperplanes in $R^m$ which intersect the line segment $\overline{{\bf O}a}$. The Gaussian random measure $\{Y(\cdot)\}$ is defined on the space of all hyperplanes in $R^m$ and the measure $\mu$ is invariant under the dual action of Euclidean motion group $M\!o(m)$.

Replacing the Gaussian random measure with an S$\alpha$S (Symmetric $\alpha$ Stable) random measure, we can easily obtain stable versions of the above examples. In this note, we will give further examples:

1. For hyperbolic space, taking as $S_a$ a self-similar set in $R^m$, we obtain stable motion on the hyperbolic space.

2. Take as $S_a$ the set of all spheres in $R^m$ of arbitrary radii which separate the origin $O$ and the point $a\in R^m$; then we obtain a self-similar S$\alpha$S random field as $\{X(a)=Y(S_a)\}$.

Along these lines, we will consider a multi-dimensional version of Bochner's subordination.

Authors

  • Shigeo TakenakaDepartment of Applied Mathematics
    Okayama University of Science
    Ridaicho 1-1
    Kita-ku 700-0005, Okayama, Japan
    e-mail

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