Spectral gap properties for random walks on homogeneous spaces: examples and consequences
Seminar Room 1, Newton Institute
AbstractCo-authors: J.-P Conze, B. Bekka, E .LePage
Let E be a homogeneous space of a Lie group G, p a finitely supported probability measure on G, such that supp(p) generates topologically G. We show that, in various situations, convolution by p has a spectral gap on some suitable functional space on E . We consider in particular Hilbert spaces and Holder spaces on E and actions by affine transformations. If G is the motion group of Euclidean space V ,we get equidistribution of the random walk on V. If G is the affine group of V,p has a stationary probability, and the projection of p on GL(V) satisfies "generic" conditions we get that the random walk satisfies Frechet's extreme law , and Sullivan's Logarithm law.
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