Attractor Properties of Non-Reversible Dynamics w.r.t.~Invariant Gibbs Measures on the Lattice

#### B. Jahnel, C. Kuelske

2016, v.22, №3, 507-535

ABSTRACT

We consider stochastic dynamics of lattice systems with finite local state space, possib
ly
at low temperature, and possibly non-reversible.
We assume the additional regularity properties on the dynamics:

a) There is at least one stationary measure which is a Gibbs measure for an absolutely s
ummable potential $\Phi$.

b) Zero loss of relative entropy density under dynamics implies the Gibbs property with
the same $\Phi$.

We prove results on the attractor property of the set of Gibbs measures for~$\Phi$:
%
%
%We consider general finite-state space models on the lattice defined in terms of transl
ation-invariant absolutely summable potentials. Further w
%
%We consider
% lattice-translation-invariant discrete-time Markovian dynamics or continuous-time inte
racting particle systems which admit at least one translation-invariant Gibbs measure $\ mu$ for some Gibbsian potential $\Phi$ as a time-stationary state.
%The dynamics may be non-reversible or reversible for this state and may also possess mo
re then one invariant measures.
%
%Under the natural further assumption that zero loss of relative entropy density w.r.t $\mu$
%%entropic loss
%already implies being a Gibbs measure w.r.t $\Phi$ we exhibit results on the attractor
property of the set of Gibbs measures for $\Phi$:

1. The set of weak limit points of any trajectory of translation-invariant measures cont
ains at least one Gibbs state for $\Phi$.
2. We show that if all elements of a weakly convergent sequence of measures are Gibbs measures for a sequence of some translation-invariant summable potentials with uniform bound, then the limiting measure must be a Gibbs measure for $\Phi$.
%for the same potential as the given time-stationary Gibbs measure.

%3. In case of the continuous-time dynamics we show that the second result holds under weaker conditions.
3. We give an extension of the second result to trajectories which are allowed to be non-Gibbs,
but have a property of asymptotic smallness of discontinuities. An example for this situation is the time evolution from a low temperature Ising measure by weakly dependent spin flips.

Keywords: Markov chain, PCA, IPS, non-equilibrium, non-reversibility, attractor property, relative entropy, Gibbsianness, non-Gibbsianness, synchronisation