Duration: 47:29

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Consider the fractional Brownian motions on the real line. What should we expect if we replace the real line by a manifold M? We will provide an answer to this question, extending work begun by Paul Levy in 1965. We will construct a family of Gaussian processes indexed by M with certain properties and argue that these objects are the proper generalization of fractional Brownian motion to the setting of GRF-s over a manifold. We also construct analogs for the Ornstein-Uhlenbeck process indexed by M. After discussing existence, invariance, self-similarity, regularity and Hausdorff dimension, we will give some examples for different manifolds, discuss simulation, and suggest some open problems for future work. 

See more on this video at https://www.microsoft.com/en-us/research/video/pacific-northwest-probability-seminar-random-self-similar-trees-dynamical-pruning-and-its-applications-to-inviscid-burgers-equations/