We study the convergence rate to equilibrium for a family of Markov semigroups
generated by a class of hypoelliptic stochastic differential equations on
,
including Galerkin truncations of the incompressible Navier–Stokes equations,
Lorenz-96, and the shell model SABRA. In the regime of vanishing, balanced
noise and dissipation, we obtain a sharp (in terms of scaling) quantitative
estimate on the exponential convergence in terms of the small parameter
. By
scaling, this regime implies corresponding optimal results both for fixed
dissipation and large noise limits or fixed noise and vanishing dissipation
limits. As part of the proof, and of independent interest, we obtain
uniform-in-
upper and lower bounds on the density of the stationary measure. Upper bounds are obtained
by a hypoelliptic Moser iteration, the lower bounds by a de Giorgi-type iteration (both
uniform in
).
The spectral gap estimate on the semigroup is obtained by a weak Poincaré
inequality argument combined with quantitative hypoelliptic regularization of the
time-dependent problem.