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.