Curvature and Other Local Inequalities in Markov Semigroups
arxiv(2024)
Abstract
Inspired by the approach of Ivanisvili and Volberg towards functional
inequalities for probability measures with strictly convex potentials, we
investigate the relationship between curvature bounds in the sense of
Bakry-Emery and local functional inequalities. We will show that not only is
the earlier approach for strictly convex potentials extendable to Markov
semigroups and simplified through use of the Γ-calculus, providing a
consolidating machinery for obtaining functional inequalities new and old in
this general setting, but that a converse also holds. Local inequalities
obtained are proven equivalent to Bakry-Emery curvature. Moreover we will
develop this technique for metric measure spaces satisfying the RCD condition,
providing a unified approach to functional and isoperimetric inequalities in
non-smooth spaces with a synthetic Ricci curvature bound. Finally, we are
interested in commutation properties for semi-group operators on ℝ^n
in the absence of positive curvature, based on a local eigenvalue criteria.
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