Self-similar Gaussian Markov processes

2020/08/07 by Bauer, Benedict, Gerhold, Stefan
#15A16 (Secondary) #39B22 #47D03 #60G15 (Primary) 60G18 #60G22 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.2008.03052

Abstract

We characterize all multi-dimensional real self-similar Gaussian Markov processes. Three types of covariance matrix functions occur: white-noise type functions, covariances that can be expressed by continuous matrix semigroups, and covariances based on non-continuous solutions of Cauchy's functional equation. Characterizing the latter requires us to develop some results on the representation theory of non-continuous matrix semigroups, which are presented in a companion paper. In dimension one, besides white noise, the self-similar Gaussian Markov processes reduce to a two-parameter family of time-changed Brownian motions. This observation simplifies several proofs of non-Markovianity of concrete processes found in the literature.

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