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