Abstract
The Fourier transform on Euclidean spaces is a powerful tool for probability analysis and the celebrated Lévy-Khinchin formula is the Fourier transform of a convolution semigroup, or the distribution of a Lévy process. On a symmetric space, a special type of homogeneous spaces, the spherical transform plays a similar role, and leads naturally to a Lévy-Khinchin type formula. The purpose of this chapter is to obtain a spherical Lévy-Khinchin formula, and use this formula together with the inverse spherical transform to prove the existence of a smooth density for a convolution semigroup on a symmetric space, and to obtain a representation for this density in terms of spherical functions. As an application, the exponential convergence to the uniform distribution is obtained in the compact case. The results will be stated in terms of convolution semigroups, and established under a nondegenerate diffusion part or an asymptotic condition on the Lévy measure.
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Liao, M. (2018). Spherical Transform and Lévy-Khinchin Formula. In: Invariant Markov Processes Under Lie Group Actions. Springer, Cham. https://doi.org/10.1007/978-3-319-92324-6_5
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DOI: https://doi.org/10.1007/978-3-319-92324-6_5
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