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For a family of infinite-dimensional diffusions with degenerate noise, wedevelop a modified $\Gamma$ calculus on finite-dimensional projections of theequation in order to produce explicit functional inequalities that can bescaled to infinite dimensions. The choice of our $\Gamma$ operator appearscanonical in our context, as the estimates depend only on the induced controldistance. We apply the general analysis to a number of examples, exploringimplications for quasi-invariance and uniqueness of stationary distributions.

Functional inequalities for a family of infinite-dimensional diffusions with degenerate noise