Stable random fields and geometry

Shigeo Takenaka

Banach Center Publications, Tome 89 (2010), p. 225-241 / Harvested from The Polish Digital Mathematics Library

Résumé

Let (M,d) be a metric space with a fixed origin O. P. Lévy defined Brownian motion X(a); a ∈ M as 0. X(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 ${Y(B); B ∈ ℬ(S^{m})}$ 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 μ(B), where μ is the uniform measure on $S^{m}$, 3. if B₁ ∩ B₂ = ∅ then Y(B₁) is independent of Y(B₂). 4. for $B_{i}$, i = 1,2,..., $B_{i} ∩ B_{j} = ∅$, i ≠ j, we have $Y(∪B_{i}) = ∑ Y(B_{i})$, a.e. Set $S_{a} = H_{a}∆H_{O}$, where $H_{a}$ is the hemisphere with center a, and ∆ means symmetric difference. Then ${X(a) = Y(S_{a}); a∈ S^{m}}$ 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 $\overline{Oa}$. The Gaussian random measure Y(·) is defined on the space of all hyperplanes in $R^{m}$ and the measure μ is invariant under the dual action of Euclidean motion group Mo(m). Replacing the Gaussian random measure with an SαS (Symmetric α 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 $a ∈ R^{m}$; then we obtain a self-similar SαS random field as ${X(a) = Y(S_{a})}$. Along these lines, we will consider a multi-dimensional version of Bochner’s subordination.

Détail

Publié le : 2010-01-01

Zbl 1210.60049

EUDML-ID : urn:eudml:doc:286229