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Additional info for Circulation Distribution, Entropy Production and Irreversibility of Denumerable Markov Chains

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N wc− n Since the state space S of ξ is ﬁnite, the state space of the derived chain {ηn }n≥0 is ﬁnite. 3, for P-almost every ω, lim n→+∞ Wn (ω) = lim n→+∞ n c∈C∞ wc log = c∈C∞ = 1 2 wc wc,n (ω) log n wc− wc wc− (wc − wc− ) log c∈C∞ wc = ep . wc− Now we discuss the ﬂuctuations of general observables. Let ϕ : S → R be n n an observable and Φn (ω) = k=0 ϕ(ξk (ω)) = k=0 ϕ(ξ0 (θk ω)). Clearly, Φn −n satisﬁes Φn (rω) = Φn (θ ω), ∀ω ∈ Ω. From the Birkhoﬀ ergodic theorem, it follows that limn→+∞ Φnn = E Π ϕ.

One can also ﬁnd its proof in Br´emaud [45, page 119]. 1. Suppose that X = {Xn }n≥0 is a homogeneous, irreducible and positive recurrent Markov chain with a countable state space S. Let µ = (µi )i∈S be the unique invariant probability distribution of X. For each i ∈ S, deﬁne Ti = inf{n ≥ 1 : Xn = i}. Then for any i, j ∈ S, i = j, the following identity holds: Prob(Tj < Ti |X0 = i) = 1 . L. Chung . We replicate it here to make the presentation more self-contained. 2. Assume that X = {Xn }n≥0 is a homogeneous Markov chain with a denumerable state space S.

Suppose that X = {Xn }n≥0 is a homogeneous, irreducible and positive recurrent Markov chain with a countable state space S. Let µ = (µi )i∈S be the unique invariant probability distribution of X. For each i ∈ S, deﬁne Ti = inf{n ≥ 1 : Xn = i}. Then for any i, j ∈ S, i = j, the following identity holds: Prob(Tj < Ti |X0 = i) = 1 . L. Chung . We replicate it here to make the presentation more self-contained. 2. Assume that X = {Xn }n≥0 is a homogeneous Markov chain with a denumerable state space S.