Little seems to be known about the sensitivity of steady states for stochastic systems. This paper discusses the problem of whether there is an invariant measure in a degenerately damped stochastic Lorenz-Stenflo model. Precisely, the solution is proved to be a nice diffusion via the Lie bracket technique and non-trivial Lyapunov functions. The finiteness of the expected positive recurrence time entails the existence problem. On the other hand, a cut-off function is constructed to show the non-existence result via proof by contradiction. For other interesting cases, the expected recurrence time is shown to be infinite.