Summary
Martingales and stochastic integrals are applied to prove Poincaré-type inequalities involving probability distributions on the Euclidean space. These inequalities generalize and improve several results in the literature and are shown to yield weighted Poincaré inequalities on some special compact manifolds. This leads to a new method of calculating all the eigenvalues and eigenfunctions of the Laplacian on then-sphere. As a by-product the eigenvalues are shown to be related to the moments of a probability distribution.
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Chen, L.H.Y. Poincaré-type inequalities via stochastic integrals. Z. Wahrscheinlichkeitstheorie verw Gebiete 69, 251–277 (1985). https://doi.org/10.1007/BF02450283
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DOI: https://doi.org/10.1007/BF02450283