Stable random fields and geometry

Shigeo Takenaka. "Stable random fields and geometry." Banach Center Publications 90.1 (2010): 225-241. <http://eudml.org/doc/286229>.

@article{ShigeoTakenaka2010,
abstract = {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.},
author = {Shigeo Takenaka},
journal = {Banach Center Publications},
keywords = {stable random fields; multi-parameter additive processes; subordination},
language = {eng},
number = {1},
pages = {225-241},
title = {Stable random fields and geometry},
url = {http://eudml.org/doc/286229},
volume = {90},
year = {2010},
}

TY - JOUR
AU - Shigeo Takenaka
TI - Stable random fields and geometry
JO - Banach Center Publications
PY - 2010
VL - 90
IS - 1
SP - 225
EP - 241
AB - 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.
LA - eng
KW - stable random fields; multi-parameter additive processes; subordination
UR - http://eudml.org/doc/286229
ER -