Summary: 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 Lévy 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 Lévy 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 develop 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.
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