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In many instances, the dynamical richness and complexity observed in naturalphenomena can be related to stochastic drives influencing their temporalevolution. For example, random noise allied to spatial asymmetries may inducestabilization of otherwise diverging trajectories in dynamical systems.However, to identify how exactly this takes place in actual processes usuallyis not a simple task. Here we unveil a few trends leading to dynamicalstabilization and diversity of behavior by introducing Gaussian white noise toa class of exactly solvable non-linear deterministic models displayingspace-dependent drifts. For the resulting nonlinear Langevin equations, theassociated Fokker-Planck equations can be solved through the similarity methodor the Fourier transform technique. By comparing the cases with and withoutnoise, we discuss the changes in the systems dynamical characteristics. Simpleexamples of drift and diffusion coefficients are explicitly analyzed andcomparisons with some other models in the literature are made. Our studyillustrates the rich phenomenology originated from spatially heterogeneousdynamical systems under the influence of white noise.

Asymmetric space-dependent systems: Partial stabilization through the  addition of noise and exact solutions for the corresponding nonlinear  Langevin equations