Dualities for the Domany-Kinzel model

We study the Domany-Kinzel model, which is a class of discrete-time Markov processes in one-dimension with two parameters (P₁, P₂) ∈ [0, 1]². When P₁ = αβ and P₂ = α(2β-β²) with (α, β) ∈ [0, 1] ², the process can be identified with the mixed site-bond oriented percolation model on a square lattice with probabilities α of a site being open and β of a bond being open. This paper treats dualities for the Domany-Kinzel model ξ_t^A and the DKdual η_t^A starting from A. We prove that (i) E(x ^{|ξtA ∩ B|}) = E(x^{|ξtB ∩ A|}) if x = 1-(2P ₁-P₂)/P₁², (ii) E(x ^{|ξtA ∩ B|}) = E(x^{|ηtB ∩ A|}) if x = 1-(2P₁-P₂)/P₁, and (iii) E(x ^{|ηtA ∩ B|}) = E(x^{|ηtB ∩ A|}) if x = 1-(2P₁-P₂), as long as one of A, B is finite and P ₂ ≤ P₁.