On Hyperboundedness and Spectrum of Markov Operators

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For reversible Markov kernels on a probability space, existence of hyperboundedness (Pt or M mapping L2 into Lp for some p>2) forces a positive spectral gap and the paper gives an explicit quantitative lower bound linking the gap to how strong the L2→Lp gain is. Proof uses finite-kernel approximation, a new intermediate connectivity spectrum between isoperimetric constants and the usual spectrum, and higher-order Cheeger type inequalities; extensions cover generators, compact manifolds, additively weighted drifts and limits as a weight parameter β grows.

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1. 0:00Intro
2. 0:13Reversible Markov operators
3. 6:45Hypercontractivity
4. 14:36A hyperboundedness estimate
5. 16:59Proof (3)
6. 18:18A contradictory argument (1)
7. 20:08A contradictory argument (2)
8. 21:18Extension of definitions
9. 23:54Markovian higher order Cheeger's inequalities
10. 25:20Dirichlet connectivity spectrum
11. 28:28Usual and Dirichlet connectivity spectra
12. 29:36Dirichlet conditions
13. 31:25Riemannian setting Let us return to the Riemannian setting of Cheeger (1970). The
14. 37:06Witten Laplacian
15. 42:13On a result of Wang
16. 43:58Changing the indices
17. 45:05A quantitative version
18. 46:38Ornstein-Uhlenbeck generator
19. 47:42A disappointing application?
20. 48:42Tensorization