By Jiang D.-Q., Qian M.

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

**Sample text**

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 ]].

3 and c− denotes the reversed cycle of c. 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 ]].

41) where q(yk , yl ) denotes the probability that the derived chain η starting at yk visits yl before returning to yk . For y1 = [i1 , i2 , · · · , is−1 ] and y2 = [i1 , i2 , · · · , is−1 , is ], we have q(y1 , y2 ) = pis−1 is , q(y2 , y1 ) = 1 − f (is , is |{i1 , i2 , · · · , is−1 }), where f (is , is |{i1 , i2 , · · · , is−1 }) denotes the probability that the original chain ξ starting at is returns to is before visiting any of the states i1 , i2 , · · · , is−1 . 42) Π s−1 s and 44 1 Denumerable Markov Chains ˜ i ([i1 , · · · , is−1 ])pi i g(is , is |{i1 , · · · , is−1 }).