Fractional Brownian Motion Overview

• Fractional Brownian motion is a self-similar Gaussian process defined by its Hurst exponent, exhibiting stationary increments and long-range dependence.
• Multiple constructions, including Lévy and Mandelbrot–van Ness methods, provide frameworks to capture either non-stationary or stationary increment properties.
• Its versatility is demonstrated in modeling anomalous transport, financial markets, and turbulent systems, highlighting its broad practical applications.

Fractional Brownian motion (fBm) is a family of zero-mean, self-similar Gaussian processes with rich temporal long-range dependence, parametrized by the Hurst exponent $H$, and notable for its central role in the modeling of anomalous diffusion, irreducibility to Markovian or martingale limits (except at $H = 1/2$), and for its versatility in representing both anti-persistent and persistent temporal correlations. fBm occurs in diverse contexts including statistical physics, finance, hydrology, turbulence, and signal processing.

1. Definitions, Characterizations, and Mathematical Structure

fBm with Hurst exponent $H \in (0,1)$ is a centered Gaussian process $\{B_H(t),\ t\in\mathbb{R}\}$ uniquely determined by two properties:

• Self-similarity: For all $c>0$, $\{B_H(ct)\} \overset{d}{=} \{c^H B_H(t)\}$.
• Stationary increments: For all $t, \tau$, the distribution of $B_H(t+\tau)-B_H(\tau)$ depends only on $t$.

The covariance function on $\mathbb{R}$ is

$$\mathrm{Cov}(B_H(s), B_H(t)) = \frac{1}{2}\left( |s|^{2H} + |t|^{2H} - |t-s|^{2H}\right).$$

The variance is $\mathbb{E}[B_H(t)^2] = |t|^{2H}$, and $$\Delta B_H = B_H(t+\tau)-B_H(t) \sim \mathcal{N}(0, |\tau|^{2H})$$.

Three closely related but formally distinct constructions exist:

• Lévy fBm (Riemann-Liouville): $x_L(t) = I_{0+}^{H+1/2}[\xi](t)$, defined on $[0,T]$, where $I_{0+}^\alpha$ is the left-sided Riemann–Liouville fractional integral, and $\xi$ is white noise. This construction yields non-stationary increments (Benichou et al., 2023).
• One-sided Mandelbrot–van Ness (MvN): Defined on $[0,\infty)$, employing integrals with lower bounds $(-\infty,0)$, producing stationary increments and the classical fBm covariance (Benichou et al., 2023).
• Two-sided MvN: Defined on $\mathbb{R}$ using both left- and right-sided fractional integrals, giving stationary increments everywhere.

All constructions yield Gaussian processes with covariance of the form above. The key distinction lies in the increment stationarity: only MvN constructions are increment-stationary for all $t$.

2. Path Integral and Action Formalisms

The sample path distribution of any zero-mean Gaussian process $x(t)$ over a domain $\mathcal{D}$ admits a path integral form

$$P[x(\cdot)] \propto \exp\left(-S[x]\right),\qquad S[x] = \frac12 \int_\mathcal{D}\int_\mathcal{D} x(t_1) K(t_1,t_2)x(t_2) dt_1dt_2,$$

where $K$ is the inverse covariance kernel: $$\int_\mathcal{D} K(t, t') \operatorname{Cov}(t', t'')dt' = \delta(t-t'')$$.

Bénichou & Oshanin (Benichou et al., 2023) showed that, for all three fBm types, the action admits a unifying representation in terms of (left- or right-sided) Riemann–Liouville fractional integrals, with the fractional order determined solely by $H$:

• For $0$$S[x] = B_H \int_\mathcal{D} [I_{\ell}^{1/2-H}\dot x(t)]^2 dt$$.
• For $1/2$$S[x] = B_H \int_\mathcal{D} [I_\ell^{3/2-H} x''(t)]^2 dt$$. The parameter $B_H$ is an explicit $H$-dependent normalization.

The only aspect distinguishing the canonical constructions is the domain and the limits of the fractional integrals, e.g., $[0,T]$ (Lévy), $[0,\infty)$ (one-sided MvN), or $(-\infty, t)$ (two-sided MvN).

3. Sample Path Properties and Regularity

Sample paths of fBm display distinctive regularity properties determined by $H$ (Zili, 2017, Ichiba et al., 2020):

• Hölder continuity: fBm paths are almost surely Hölder continuous of any order $\alpha < H$ but of no higher order.
• Nowhere differentiability: For $H \leq 1$, fBm paths are almost surely nowhere differentiable:

$$\limsup_{t\to t_0} \left| \frac{B_H(t)-B_H(t_0)}{t-t_0}\right| = +\infty,\qquad \forall t_0.$$

For generalized fBm with sufficiently large regularity parameter $\alpha > 1/2$, one obtains differentiability in mean square but not twice differentiable (Ichiba et al., 2020).

• Fractal dimension: The graph $t \mapsto B_H(t)$ has Hausdorff dimension $D=2-H$ (Lilly et al., 2016).

For $H < 1/2$, increments are anti-persistent (negative correlations), leading to "rougher" sample paths, while $H > 1/2$ gives persistent, smoother but still non-differentiable trajectories.

4. Covariance, Spectral Structure, and Long-Range Dependence

The fBm process is characterized by stationary increments and long-range dependence. Key features include:

• Covariance structure:

$$\mathbb{E}[B_H(s)B_H(t)] = \frac12 \left(|s|^{2H}+|t|^{2H}-|t-s|^{2H}\right).$$

• Increment covariance: For large lag $k$,

$$\mathrm{Cov}(B_H(n+1)-B_H(n), B_H(n+k+1)-B_H(n+k)) \sim H(2H-1)k^{2H-2},$$

so increments are long-range dependent ($\sum k^{2H-2} = \infty$) if and only if $H > 1/2$ (Zili, 2017, Mliki et al., 2021).

• Spectral density: For $H \in (0,1)$, the fBm power spectrum at high frequency decays as $|\omega|^{-2H-1}$ (Lilly et al., 2016). Steeper spectral slopes (large $H$) correspond to fewer high-frequency components and smoother paths.

5. Generalizations and Variants

Numerous extensions of the fBm construction exist:

Generalized and mixed fBm: Processes combining distinct fBm with differing parameters (e.g., coefficient-weighted positive and negative times, mixtures with Brownian motion) yield processes with lack of increment stationarity and/or lack of self-similarity, yet inherit long-range dependence and other features (Zili, 2017, Mliki et al., 2021).

fBm with fluctuating diffusivity: Replacing the diffusion constant by a random process $D(t)$ gives rise to models with rich aging and non-ergodicity (Pacheco-Pozo et al., 6 May 2024). The mean squared displacement (MSD) becomes an integral over $\langle D(u)\rangle (t-u)^{2H-1}$.

Random Hurst exponent: Allowing $H$ itself to be random (chosen per trajectory) leads to models combining anomalous diffusion and superstatistical effects, giving rise to accelerating diffusion and persistence transitions in the two-point autocovariance (Balcerek et al., 2022).

Extension to $H < 0$: Traditional fBm is undefined for $H\leq0$ due to divergence of the variance at finite $t$; however, local time-averaging regularizations yield stationary processes with finite variance and complete arrest of diffusion—a regime interpreted as "strong anti-persistence" (Meerson et al., 8 Jul 2025).

Matérn process: This can be interpreted as a damped fBm with spectral density $S(\omega) = A^2 / (\omega^2 + \lambda^2)^{\alpha}$, exhibiting an fBm-type scaling at high frequencies and a plateau in the low-frequency region, resulting in normal diffusive behavior for the integrated process (Lilly et al., 2016).

6. Statistical Behavior, Extreme Values, and Local Time

• Extreme-value statistics: For $H\neq 1/2$, the distributions for the maximum and the time at which it is achieved deviate from the classical Arcsine and Gaussian laws. A perturbative expansion for $H=1/2+\varepsilon$ yields explicit corrections, capturing the non-Markovian fingerprint: as the scaling exponent $H$ departs from $1/2$, the distributions interpolate smoothly, and the persistence exponent becomes $\theta=1-H$ (Delorme et al., 2016, Delorme et al., 2015).
• Local time: The occupation measure (local time) exists for each $H<1$, is square-integrable, and may be pathwise constructed via normalized level crossing counts in Lebesgue partitions, reflecting the deep interplay between long-range dependence and the non-Markovian nature of the process (Das et al., 2023).
• Variation along random partitions: The $(1/H)$-variation along level-crossing–based partitions converges almost surely to a constant $\mathfrak{c}_H t$, with $\mathfrak{c}_H$ encoding non-Markovian effects, distinct from Brownian motion’s value (Das et al., 2023).

7. Applications, Simulation, and Modeling

fBm provides a foundational toolset in modeling:

• Anomalous transport: Modeling sub- and superdiffusive transport in biological, soft condensed matter and microfluidic systems. When bounded by reflecting boundaries, non-uniform, non-Gaussian stationary distributions (accretion/depletion near boundaries) emerge, with marked effects on reaction kinetics near interfaces (Guggenberger et al., 2019, Wada et al., 2017).
• Financial mathematics: fBm-driven market models encode long-memory effects and explicit decomposition into "martingale-noise" and "smooth/predictable" components, enabling optimal mean–variance portfolio design even in the absence of the semimartingale property (Dokuchaev, 2015).
• Turbulence and environmental science: In turbulent dispersion, fBm and its Matérn generalization effectively model trajectory velocities, with proper spectrum and realistic diffusive properties at large times (Lilly et al., 2016).
• Simulation: Exact generation relies on circulant-embedding/Davis–Harte methods for regular fBm, or $O(N\log N)$ FFT-based convolution approximations for damped (Matérn) variants (Lilly et al., 2016, Wada et al., 2017).
• Partial differential equations: The fundamental solution for the fBm transition density satisfies a generalized diffusion equation with a time-dependent, nonlinear fractional differential equation for the diffusivity, encoding the anomalous scaling (Garra et al., 2018).

Table: Main Constructions of fBm

Construction	Domain	Covariance Function	Increment Stationarity
Lévy fBm	$[0,T]$	$$t_2^{H+3/2} {}_2F_1(1,2-H;H+1; t_2/t_1) (t_1 t_2)^{1/2-H}$$	No
One-sided MvN	$[0,\infty)$	$t_1^{2H}+t_2^{2H}-|t_1-t_2|^{2H}$	Yes
Two-sided MvN	$\mathbb{R}$	$\frac12 (|t_1|^{2H} + |t_2|^{2H} - |t_1-t_2|^{2H})$	Yes

Outlook

Fractional Brownian motion continues to be a foundational object in the theory and modeling of self-similar, long-range dependent processes. Its unification via fractional calculus and Gaussian actions provides a robust framework for analysis and simulation (Benichou et al., 2023). Ongoing research explores extensions to non-stationary, heterogeneous, and multifractal regimes, integration with random environments (fluctuating diffusivity, random Hurst exponents), and further connections to nonlocal PDEs, ergodic theory, and ergodicity breaking (Balcerek et al., 2022, Pacheco-Pozo et al., 6 May 2024, Meerson et al., 8 Jul 2025).