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Ergodicity for infinite particle systems with locally conserved quantities. (English) Zbl 1268.60120
The paper analyses degenerate infinite-dimensional sub-elliptic generators, and obtains estimates on the long-term behavior of the corresponding Markov semigroups that describe a model of heat conduction. The authors also establish ergodicity of the system for a family of invariant measures, and show that the optimal rate of convergence to the equilibrium state is polynomial. Also, the Liggett-Nash-type inequality appears to be valid. Section 2 introduces the basic notation, and states an infinite system of stochastic differential equations of interest. The existence of a mild solution is considered in Section 3, whereas Section 4 discusses general properties of the corresponding semigroup, in particular, the existence of a family of invariant measures, strong continuity, positivity, and contractivity properties in $$L^2$$-spaces. Section 5 provides a certain characterisation of invariant sub-spaces (of Sobolev type). Ergodicity with optimal rate of convergence to the equilibrium state is demonstrated in Section 6. Section 7 derives the Liggett-Nash-type inequalities on the basis of the obtained results. A generalized dynamics of the related type, complemented with some extra features, is the topic of Section 8. It deals with the phase transitions in the stochastic paradigm. The paper also shows that, in the considered families, one can observe a change in the behavior of the decay to the equilibrium state from exponential to algebraic. Further application of the ergodicity results are in the focus of Section 9. The paper includes the detailed proofs of the aforementioned results.

##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 60H30 Applications of stochastic analysis (to PDEs, etc.) 82C22 Interacting particle systems in time-dependent statistical mechanics
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