Some results on Markov processes of infinite lattice spin systems

Abstract

Lem m a 2. 1. p o f .9 is a limiting Gibbsian measure if and only i f (2. 4) holds only f or any one point set V = { x } , x EZ ".

Ergodic and mixing properties of equilibrium measures for Markov processes

“…To see that the hypothesis of part i) of Theorem (1.4) is fulfilled too, we appeal to Proposition (4.1) and to the fact that all weak limits of pS(t) (p G AA7) belong to " §. This last statement was proved by Holley [10] for finite range potentials, and extended to potentials satisfying (5.1) by Higuchi and Shiga [9]. These considerations show that all elements of (AÂA?…”

Attractor Properties for Irreversible and Reversible Interacting Particle Systems

“…We are looking for monotonicity in suitably chosen finite-volume approximations of g L to conclude that g L is upper semicontinuous. This is what is done in [16], making heavy use of reversibility which we cannot use in general, and taking advantage of the two-spin situation. We work with a useful decomposition of g L to treat also irreversible dynamics, see (5) and (21).…”

Some problems related to Gibbs states, canonical Gibbs states and Markovian time evolutions

“…Thus, we have de(x,4)= 1 for all x, because/zoo is everywhere dense by Lemma 1.1. Hence (b is trivial.…”

“…The following theorem can be proved in a similar way as in the case of an infinite lattice spin system [4], [6]. The proof is essentially same as Lemma 2.7 in [4]. However, we use the symmetry of p(x, y) and the following formula;…”