Minnesota Journal of Undergraduate Mathematics

Minimal Noise-Induced Stabilization of One-Dimensional Diffusions

Tony Allen

West Virginia University

Emily Gebhardt

Adam Kluball

Tiffany Kolba

Valparaiso University

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Abstract

The phenomenon of noise-induced stabilization occurs when an unstable deterministic system of ordinary differential equations is stabilized by the addition of randomness into the system. In this paper, we investigate under what conditions one-dimensional, autonomous stochastic differential equations are stable, where we take the notion of stability to be that of global stochastic boundedness. Specifically, we find the minimum amount of noise necessary for noise-induced stabilization to occur when the drift and noise coefficients are power, polynomial, exponential, or logarithmic functions.