JournalsrmiVol. 19, No. 3pp. 767–796

Elliptic Self-Similar Stochastic Processes

  • Albert Benassi

    Université Blaise Pascal, Aubière, France
  • Daniel Roux

    Université Blaise Pascal, Aubière, France
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Abstract

Let MM be a random measure and LL be an elliptic pseudo-differential operator on Rd\mathbb{R}^d. We study the solution of the stochastic problem LX=MLX=M, X(0)=0X(0)=0 when some homogeneity and integrability conditions are assumed. If MM is a Gaussian measure the process XX belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of MM is not necessarily Gaussian. We characterize the solutions XX which are self-similar and with stationary increments in terms of the driving measure MM. 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.

Cite this article

Albert Benassi, Daniel Roux, Elliptic Self-Similar Stochastic Processes. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 767–796

DOI 10.4171/RMI/369