By Jiang D.-Q., Qian M.

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

**Example text**

L. Chung [62]. 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. For any H ⊂ S, i, j ∈ S and n ∈ N, deﬁne the taboo probability p(i, j, n|H) = Prob(Xn = j, Xm ∈ H for 1 ≤ m < n|X0 = i). e. j can be reached from i under the taboo H), then lim N →+∞ 1+ 1+ N n=1 p(j, j, n|H) N n=1 p(i, i, n|H) = = +∞ n=1 p(i, j, n|H ∪ {i}) +∞ n=1 p(i, j, n|H ∪ {j}) N 1 + n=1 p(j, j, n|H ∪ {i}) lim . 7. 1 to the stationary Markov chain ξ, then we can obtain πi P(Tj < Ti |ξ0 = i) = πj P(Ti < Tj |ξ0 = j), ∀i, j ∈ S, i = j, which together with g(j, j|{i}) = [1 − P(Tj < Ti |ξ0 = j)]−1 = [P(Ti < Tj |ξ0 = j)]−1 implies the following identity: πi g(j, j|{i}) = πj g(i, i|{j}), ∀i, j ∈ S, i = j.

Clearly, Φn −n satisﬁes Φn (rω) = Φn (θ ω), ∀ω ∈ Ω. From the Birkhoﬀ ergodic theorem, it follows that limn→+∞ Φnn = E Π ϕ. Use the Perron-Frobenius theorem, then one sees that 1 def c(λ1 , λ2 ) = lim log Eeλ1 Wn +λ2 Φn n→+∞ n exists and is diﬀerentiable with respect to λ1 , λ2 . Thus {µn : n > 0}, the family of the distributions of {( Wnn , Φnn ) : n > 0}, has a large deviation property with rate function I(z1 , z2 ) = supλ1 ,λ2 ∈R {λ1 z1 +λ2 z2 −c(λ1 , λ2 )}. It is not diﬃcult to ﬁnd that c(λ1 , λ2 ) = c(−(1 + λ1 ), λ2 ) and I(z1 , z2 ) = I(−z1 , z2 ) − z1 .

E. j can be reached from i under the taboo H), then lim N →+∞ 1+ 1+ N n=1 p(j, j, n|H) N n=1 p(i, i, n|H) = = +∞ n=1 p(i, j, n|H ∪ {i}) +∞ n=1 p(i, j, n|H ∪ {j}) N 1 + n=1 p(j, j, n|H ∪ {i}) lim . 7. 1 to the stationary Markov chain ξ, then we can obtain πi P(Tj < Ti |ξ0 = i) = πj P(Ti < Tj |ξ0 = j), ∀i, j ∈ S, i = j, which together with g(j, j|{i}) = [1 − P(Tj < Ti |ξ0 = j)]−1 = [P(Ti < Tj |ξ0 = j)]−1 implies the following identity: πi g(j, j|{i}) = πj g(i, i|{j}), ∀i, j ∈ S, i = j. 41) where q(yk , yl ) denotes the probability that the derived chain η starting at yk visits yl before returning to yk .

### Circulation Distribution, Entropy Production and Irreversibility of Denumerable Markov Chains by Jiang D.-Q., Qian M.

by Ronald

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