Invariant distributions: Applications in the real World

Question

I'm studying about invariant distributions for Markov processes; say in the context of dynamics of Random Neural Networks (biological Networks).

I can't fully understand what does an invariant distribution say about the actual phisical phenomenon ?

From what I understand, we know that the distribution of the process will tend to the invariant distribution, so after "a while" said distribution and the invariant one should be alike...?

Answer 1

As an example: Monte Carlo Markov chain: https://en.wikipedia.org/wiki/Markov_chain_Monte_Carlo