Self-similar Gaussian Markov processes
Probability
2025-08-13 v2
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.
Keywords
Cite
@article{arxiv.2008.03052,
title = {Self-similar Gaussian Markov processes},
author = {Benedict Bauer and Stefan Gerhold},
journal= {arXiv preprint arXiv:2008.03052},
year = {2025}
}