Sensitivity of steady states in a degenerately-damped stochastic Lorenz system

TL;DR

This paper investigates the stability and invariant measures of a stochastic Lorenz system with degenerately applied damping, revealing conditions under which the system admits a unique invariant measure or none at all.

Contribution

It provides new theoretical results on the existence and uniqueness of invariant measures for a degenerate stochastic Lorenz system, using Lyapunov functions.

Findings

Unique invariant measure when noise acts on the convection variable without damping in temperature.

No normalizable invariant state when a positive growth term is present in the temperature profile.

Lyapunov function analysis determines recurrence or transience of the system.

Abstract

We study stability of solutions for a randomly driven and degenerately damped version of the Lorenz '63 model. Specifically, we prove that when damping is absent in one of the temperature components, the system possesses a unique invariant probability measure if and only if noise acts on the convection variable. On the other hand, if there is a positive growth term on the vertical temperature profile, we prove that there is no normalizable invariant state. Our approach relies on the derivation and analysis of non-trivial Lyapunov functions which ensure positive recurrence or null-recurrence/transience of the dynamics.