Elliptic Self-Similar Stochastic Processes

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.