  For a family of infinite-dimensional diffusions with degenerate noise, we
develop a modified $\Gamma$ calculus on finite-dimensional projections of the
equation in order to produce explicit functional inequalities that can be
scaled to infinite dimensions. The choice of our $\Gamma$ operator appears
canonical in our context, as the estimates depend only on the induced control
distance. We apply the general analysis to a number of examples, exploring
implications for quasi-invariance and uniqueness of stationary distributions.
