  Fractional Brownian motion (fBm) is an experimentally-relevant, non-Markovian
Gaussian stochastic process with long-ranged correlations between the
increments, parametrised by the so-called Hurst exponent $H$; depending on its
value the process can be sub-diffusive $(0 < H < 1/2)$, diffusive $(H = 1/2)$
or super-diffusive $(1/2 < H < 1)$. There exist three alternative equally often
used definitions of fBm -- due to L\'evy and due to Mandelbrot and van Ness
(MvN), which differ by the interval on which the time variable is formally
defined. Respectively, the covariance functions of these fBms have different
functional forms. Moreover, the MvN fBms have stationary increments, while for
the L\'evy fBm this is not the case. One may therefore be tempted to conclude
that these are, in fact, different processes which only accidentally bear the
same name. Recently determined explicit path integral representations also
appear to have very different functional forms, which only reinforces the
latter conclusion. Here we develope a unifying equivalent path integral
representation of all three fBms in terms of Riemann-Liouville fractional
integrals, which links the fBms and proves that they indeed belong to the same
family. We show that the action in such a representation involves the
fractional integral of the same form and order (dependent on whether $H < 1/2$
or $H > 1/2$) for all three cases, and differs only by the integration limits.
