Consider a sequence of continuous-time Markov chains $(X^{ (n)}_t:t\ge 0)$ evolving on a fixed finite state space $V$. Let $I_n$ be the level two large deviations rate functional for $X^{ (n)}_t$, as $t\to\infty$. Under a hypothesis on the jump rates, we prove that $I_n$ can be written as $I_n = I^{ (0)} \,+\, \sum_{ 1\le p\le q} (1/\theta^{ (p)}_n) \, I^{ (p)}$ for some rate functionals $I^{ (p)}$. The weights $\theta^{ (p)}_n$ correspond to the time-scales at which the sequence of Markov chains $X^{ (n)}_t$ exhibit a metastable behavior, and the zero level sets of the rate functionals $I^{ (p)}$ identify the metastable states.
Categories: math.PR