Introduction to Analysis on the Discrete Cube III
summary
Compares concrete semigroups and computes generators explicitly: shift group has generator −d/dx and fails analytic recovery for smooth nonanalytic examples like e^(−1/x^2); Brownian motion yields the heat semigroup with generator −½Δ via Gaussian convolution and integration by parts; Ornstein–Uhlenbeck gives generator x·∇−Δ and has Gaussian invariant measure. On the discrete cube uses Walsh Fourier, CLT and triangle inequalities to get spectral gap=1 (even:1/2), Stroock–Varopoulos bounds, Khinchin/Kahane constants (L2 vs L1 = √2, L4 vs L2 = 3^(1/4)), and Bonami lemma E[X^4] ≤ 9^d E[X^2]^2 for degree d chaos.
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chapters (6)
- 0:00Start
- 46:26Poincare inequality for even functions
- 49:01Poincare inequality - proof of the second assertion
- 50:22Spectral gap
- 52:11Elementary inequality - end of the proof Recall
- 55:24Stroock-Varopoulos inequality - end of the proof Thus (0) 20. On the other hand
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