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December, 2003 Elliptic Self Similar Stochastic Processes
Albert Benassi, Daniel Roux
Rev. Mat. Iberoamericana 19(3): 767-796 (December, 2003).

Abstract

Let $M$ be a random measure and $L$ be an elliptic pseudo-differential operator on $\mathbb{R}^d$. We study the solution of the stochastic problem $LX=M$, $X(0)=0$ when some homogeneity and integrability conditions are assumed. If $M$ is a Gaussian measure the process $X$ belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of $M$ is not necessarily Gaussian. We characterize the solutions $X$ which are self-similar and with stationary increments in terms of the driving measure $M$. Then we use appropriate wavelet bases to expand these solutions and we give regularity results. In the last section it is shown how a percolation forest can help with constructing a self-similar Elliptic Process with non stable law.

Citation

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Albert Benassi. Daniel Roux. "Elliptic Self Similar Stochastic Processes." Rev. Mat. Iberoamericana 19 (3) 767 - 796, December, 2003.

Information

Published: December, 2003
First available in Project Euclid: 20 February 2004

zbMATH: 1055.60037
MathSciNet: MR2053563

Subjects:
Primary: ‎42C40 , 60G18
Secondary: 60G20

Keywords: elliptic processes , elliptic pseudo-differential operator , Intermittency , percolation tree , regularity of sample paths , Self-similar processes with stationary increments , wavelet basis

Rights: Copyright © 2003 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.19 • No. 3 • December, 2003
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