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By Jiang D.-Q., Qian M.

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

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If for each n > 0, Φn (rω) = Φn (θ−n ω) and Ψn (rω) = −Ψn (θ−n ω), ∀ω ∈ Ω, then it holds that c(λ, β, γ) = c(−(1 + λ), β, −γ), I(z, u, v) = I(−z, u, −v) − z. Proof. For any given λ, β, γ, EeλWn + β,Φn + γ,Ψn λ = dP[0,n] (ω) dP− [0,n] = dP[0,n] (rω) dP− [0,n] = dP[0,n] −n (θ ω) dP− [0,n] = Ee−(1+λ)Wn + e β,Φn (ω) + γ,Ψn (ω) dP(ω) λ e β,Φn (rω) + γ,Ψn (rω) dP− (ω) −λ e β,Φn (θ −n ω) + γ,−Ψn (θ −n ω) β,Φn + −γ,Ψn dP− (ω) . The desired result follows immediately. 10 (see [254]). E. Harris [219]. One can also find its proof in Br´emaud [45, page 119].

It is not difficult to find that c(λ1 , λ2 ) = c(−(1 + λ1 ), λ2 ) and I(z1 , z2 ) = I(−z1 , z2 ) − z1 . In general, let {Φn : n > 0} and {Ψn : n > 0} be two sets of random vectors on (Ω, F, P), where Φn and Ψn are F0n -measurable. Provided the free energy function def c(λ, β, γ) = lim n→+∞ 1 log EeλWn + n β,Φn + γ,Ψn exists and is differentiable, it holds that {µn : n > 0}, the family of the distributions of { n1 (Wn , Φn , Ψn ) : n > 0}, has a large deviation property with rate function 42 1 Denumerable Markov Chains λz + β, u + γ, v − c(λ, β, γ) .

Proof. For each trajectory ω of the Markov chain ξ, in Sect. 2 we defined the derived chain {ηn (ω)}n≥0 . Recall that if the length ln+1 (ω) of ηn+1 (ω) is less than the length ln (ω) of ηn (ω), then ω completes a cycle at time n + 1; if ln+1 (ω) = ln (ω), then ξn+1 (ω) = ξn (ω). We define inductively a sequence of random variables {fn (ω) : n ≥ 0} as below: def 1) f0 (ω) = 1; 2) For each n ≥ 0,  pξ (ω)ξn+1 (ω)  fn (ω) pξn (ω)ξ , def n (ω) n+1 fn+1 (ω) =  fn (ω) pi1 i2 ···pis−1 is pis i1 pi i ···pi i pi i s s−1 2 1 1 s if ln+1 (ω) ≥ ln (ω), −1 , if ηn (ω) = [ηn+1 (ω), [i1 , · · · , is ]].

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