Im trying to gain an intuitive notion of local ergodicity but I've only been able to find a quite technical post on Proof of Wiener's local ergodic theorem. I will try to ask the question in plain lingo, but I can try to be more exact if needed.
Scenario:
Image observing i...N stochastic processes that we assume are identical and driven by the same unknown dynamic which results in one-dimensional states that exists on the real number line. Over time we observe that the states only takes values in a closed interval that is a subset of the real number line. We measure the system until time $T_i$ where the ith process enters an absorbing state.
Questions:
1) Assume that for infinite time, the N processes persist on this subset. E.g. we can subscribe probability zero to the rest of the real number line. Is this (heuristically) a ergodic process we're observing?
2) Assume that 1) is true; e.g. the processes persists on the subset. Further, assume that for some very long time ($T_i\rightarrow\infty$) of one system, and a very large ensemble ($N\rightarrow\infty$), the time average equals the ensemble average. E.g. the N processes are still not mixing with the real number line, but they seem to withhold a (heuristic) notion of ergodicity (e.g. time-average equals ensemble-average). Is this somehow linked to local ergodicity?
My interest comes from trying to understand this paper: Statistical mechanics of complex neural systems and high dimensional data. Especially, I am interesting in understanding the statement (p. 9):
In the large N limit, free energy barriers between valleys diverge, so that in dynamical versions of this model, if an activity pattern starts in one valley, it will stay in that valley for infinite time. Thus ergodicity is broken, as time average activity patterns are not equal to the full Gibbs average activity pattern. The network can thus maintain multiple steady states, and we are interested in understanding the structure of these steady states.
I am assuming that we would expect behaviour that aligns to my scenario from a network that is in a steady state. E.g. stays "stable" on a subset, but does not visit any other states on the real number line.
I realise the above does not contain any math, but that is fully on purpose as I am interested in (possibly heuristic) intuition and to avoid formalism. Let me know if I should clarify my very simple example.