Revista Matemática Iberoamericana

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Volume 19, Issue 3, 2003, pp. 767–796
DOI: 10.4171/RMI/369

Published online: 2003-12-31

Elliptic Self-Similar Stochastic Processes

Albert Benassi[1] and Daniel Roux[2]

(1) Université Blaise Pascal, Aubière, France
(2) Université Blaise Pascal, Aubière, France

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.

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

Benassi Albert, Roux Daniel: Elliptic Self-Similar Stochastic Processes. Rev. Mat. Iberoam. 19 (2003), 767-796. doi: 10.4171/RMI/369