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We analyse certain degenerate infinite dimensional sub-elliptic generators,and obtain estimates on the long-time behaviour of the corresponding Markovsemigroups that describe a certain model of heat conduction. In particular, weestablish ergodicity of the system for a family of invariant measures, and showthat the optimal rate of convergence to equilibrium is polynomial.Consequently, there is no spectral gap, but a Liggett-Nash type inequality isshown to hold.

ERGODICITY FOR INFINITE PARTICLE SYSTEMS WITH LOCALLY CONSERVED QUANTITIES