Transition from ergodic to explosive behavior in a family of stochastic differential equations

TL;DR

This paper investigates a family of quadratic stochastic differential equations, identifying a critical parameter that determines whether solutions are ergodic or explode, with implications for turbulent particle transport.

Contribution

It introduces a critical parameter value in quadratic SDEs that marks the transition from ergodic to explosive behavior, combining Lyapunov functions, hypoellipticity, and control theory.

Findings

Existence of a critical parameter $\alpha_{1}=\alpha_{2}$ for system behavior

System is ergodic when $\alpha_{2}>\alpha_{1}$

Solutions explode or are not defined for all times when $\alpha_{2}<\alpha_{1}$

Abstract

We study a family of quadratic stochastic differential equations in the plane, motivated by applications to turbulent transport of heavy particles. Using Lyapunov functions, we find a critical parameter value $\alpha_{1}=\alpha_{2}$ such that when $\alpha_{2}>\alpha_{1}$ the system is ergodic and when $\alpha_{2}<\alpha_{1}$ solutions are not defined for all times. H\"{o}rmander's hypoellipticity theorem and geometric control theory are also utilized.