On perturbations of an ODE with non-Lipschitz coefficients by a small self-similar noise

Abstract

We study the limit behavior of differential equations with non-Lipschitz coefficients that are perturbed by a small self-similar noise. It is proved that the limiting process is equal to the maximal solution or minimal solution with certain probabilities $p_+$ and $p_-=1-p_+$, respectively. We propose a space-time transformation that reduces the investigation of the original problem to the study of the exact growth rate of a solution to a certain SDE with self-similar noise. This problem is interesting in itself. Moreover, the probabilities $p_+$ and $p_-$ coincide with probabilities that the solution of the transformed equation converges to $+\infty$ or $-\infty$ as $t\to\infty,$ respectively.