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. Multidimensional generalizations are also considered.
Results are based on the joint research with F. N. Proske.
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