Abstract
Consider a continuous time particle system ηt = (ηt(k), k ∈ 𝕃), indexed by a lattice 𝕃 which will be either ℤ, ℤ∕nℤ, a segment {1, ⋯ , n}, or ℤd, and taking its values in the set Eκ𝕃 where Eκ = {0, ⋯ , κ − 1} for some fixed κ ∈{∞, 2, 3, ⋯ }. Assume that the Markovian evolution of the particle system (PS) is driven by some translation invariant local dynamics with bounded range, encoded by a jump rate matrix ⊤. These are standard settings, satisfied by the TASEP, the voter models, the contact processes. The aim of this paper is to provide some sufficient and/or necessary conditions on the matrix ⊤ so that this Markov process admits some simple invariant distribution, as a product measure (if 𝕃 is any of the spaces mentioned above), the law of a Markov process indexed by ℤ or [1, n] ∩ ℤ (if 𝕃 = ℤ or {1, …, n}), or a Gibbs measure if 𝕃 = ℤ/nℤ. Multiple applications follow: efficient ways to find invariant Markov laws for a given jump rate matrix or to prove that none exists. The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution (for any memory m). (As usual, a random process X indexed by ℤ or ℕ is said to be a Markov chain with memory m ∈ {0, 1, 2, ⋯ } if ℙ(Xk ∈ A | Xk−i, i ≥ 1) = ℙ(Xk ∈ A | Xk−i, 1 ≤ i ≤ m), for any k.) We also prove that some models close to these models do. We exhibit PS admitting hidden Markov chains as invariant distribution and design many PS on ℤ2, with jump rates indexed by 2 × 2 squares, admitting product invariant measures.
Highlights
Some notationWe let N = Z+ = {0, 1, 2, . . .} and N = N \ {0}
The voter models and the contact processes are shown not to possess any Markov laws as invariant distribution. (As usual, a random process X indexed by Z or N is said to be a Markov chain with memory m ∈ {0, 1, 2, · · · } if P(Xk ∈ A |Xk−i, i ≥ 1) = P(Xk ∈ A |Xk−i, 1 ≤ i ≤ m), for any k.) We prove that some models close to these models do
For the zero range type processes we prove that there exists a family of distributions F, such that depending on T, either all the product measures ρZ are invariant by T for all ρ ∈ F, or none of them is invariant by T
Summary
For −∞ ≤ a ≤ b ≤ +∞, define a, b := [a, b] ∩ Z as the set of integers in [a, b] We will call such a set a Z-interval. If I is a Z-interval, for example I = 3, 6 , x(I) = (x3, x4, x5, x6), and we will often write x 3, 6 instead. If E is a set and I a subset of Z, or a sequence in Z, we denote by EI := {x(I) : the entries in x(I) belong to E}. For y = x(I), a sequence indexed by a set I, and for A ⊂ Z, set yA = x(I \ A), the word obtained by suppressing the letters in position belonging to A in y. A function g : A → R is said to be equivalent to 0, we write g ≡ 0, if its image is reduced to 0
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