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

TL;DR

This paper investigates how small self-similar noise affects differential equations with non-Lipschitz coefficients, revealing that the solutions tend to either maximal or minimal solutions with certain probabilities, and introduces a transformation to analyze growth rates.

Contribution

It introduces a space-time transformation that simplifies the analysis of the perturbed ODEs and characterizes the limiting behavior and probabilities of solution convergence.

Findings

The limiting process is either the maximal or minimal solution with specific probabilities.

A space-time transformation reduces the problem to analyzing an SDE with self-similar noise.

Probabilities of convergence to infinity are explicitly characterized.

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.