Fractional Brownian motion with negative Hurst exponent

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Abstract
Fractional Brownian motion (fBm) is an important scale-invariant Gaussian non-Markovian process with stationary increments, which serves as a prototypical example of a system with long-range temporal correlations and anomalous diffusion. The fBm is traditionally defined for the Hurst exponent H in the range 0 < H < 1 . Here we extend this definition to the regime − 1 / 2 < H < 0 . The extended fBm is not a pointwise process, so we regularize it via a local temporal averaging with a narrow kernel. The resulting process is both very rough and persistent, that is long-range positively correlated. In addition, this process is stationary. The stationarity implies that diffusion is completely suppressed in this region of H . We also study another closely related Gaussian process: the stationary fractional Ornstein–Uhlenbeck (fOU) process, extended to the range − 1 / 2 < H < 0 and smoothed in the same way as the fBm. Remarkably, the smoothed fOU process is asymptotically insensitive to the strength of the confining potential. Finally, we determine the optimal paths of the fBm and fOU processes for − 1 / 2 < H < 0 , conditioned on reaching a specified value when starting from zero. In the marginal case H = 0 , our results match continuously with known results for the traditionally defined fBm and fOU processes.

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