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## Banach Center Publications

2010 | 90 | 1 | 225-241
Tytuł artykułu

### Stable random fields and geometry

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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.
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Rocznik
Tom
Numer
Strony
225-241
Opis fizyczny
Daty
wydano
2010
Twórcy
autor
• Department of Applied Mathematics, Okayama University of Science, Ridaicho 1-1, Kita-ku 700-0005, Okayama, Japan
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