Fractional Brownian Motion with Negative Hurst Exponent

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 $-1/2<H<0$. Here we extend this definition to the strongly anti-persistent 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 filter. The extended fBm turns out to be stationary, and we derive its autocorrelation function. The stationarity implies a complete arrest of diffusion in this region of $H$. We also determine the variance of a 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 turns out to be insensitive to the strength of the confining potential. Finally, we determine the optimal paths of the conditioned fBm and fOU processes for $-1/2<H<0$. In the marginal case $H=0$, our results match continuously with known results for the traditionally defined fBm and fOU processes.