Let be a random measure and be an elliptic pseudo-differential operator on . We study the solution of the stochastic problem , when some homogeneity and integrability conditions are assumed. If is a Gaussian measure the process belongs to the class of Elliptic Gaussian Processes which has already been studied. Here the law of is not necessarily Gaussian. We characterize the solutions which are self-similar and with stationary increments in terms of the driving measure . 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.
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Albert Benassi, Daniel Roux, Elliptic Self-Similar Stochastic Processes. Rev. Mat. Iberoam. 19 (2003), no. 3, pp. 767–796DOI 10.4171/RMI/369