Fractional Brownian Motion with Negative Hurst Exponent

TL;DR

This paper extends the definition of fractional Brownian motion and related processes to negative Hurst exponents, revealing new properties like stationarity and suppressed diffusion, with implications for understanding long-range correlations.

Contribution

It introduces a regularization method for fBm with negative H, analyzes its properties, and explores related Gaussian processes, expanding the theoretical framework of anomalous diffusion models.

Findings

Extended fBm is stationary and highly rough for -1/2<H<0.

The smoothed fOU process is insensitive to confining potential strength.

Optimal paths for the extended processes are characterized for negative H.

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 filter. 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.