  We define a two-parameter family of Gaussian Markov processes, which includes
Brownian motion as a special case. Our main result is that any centered
self-similar Gaussian Markov process is a constant multiple of a process from
this family. This yields short and easy proofs of some non-Markovianity results
concerning variants of fractional Brownian motion (most of which are known). In
the proof of our main theorem, we use some properties of additive functions,
i.e. solutions of Cauchy's functional equation. In an appendix, we show that a
certain self-similar Gaussian process with asymptotically stationary increments
is not a semimartingale.
