Riemann-Liouville fBm

• Riemann-Liouville fBm is defined as the fractional integral of standard Brownian motion, producing a centered Gaussian process with non-stationary increments.
• Its covariance structure and self-similarity, governed by the Hurst index, underlie long-range dependence and anomalous diffusion properties.
• The process supports advanced stochastic calculus and is applied in modeling anomalous diffusion, SPDEs, and systems with random Hurst parameters.

Riemann-Liouville fractional Brownian motion (RL-fBm) is a non-Markovian centered Gaussian process defined through fractional calculus, distinguished by long-range dependence, anomalous diffusion, and non-stationary increments. Its construction employs Riemann–Liouville fractional integrals acting on standard Brownian motion, yielding a family of processes parameterized by the Hurst index $H \in (0,1)$, and serving as the canonical finite-interval (Lévy) representative among broader fBm constructions. RL-fBm admits a unified path-integral and stochastic calculus framework, with recent generalizations to random Hurst exponents and stochastic integration in infinite-dimensional settings. Its invariant properties, covariance kernel, ergodicity structure, and intrinsic connection to fractional differential equations distinguish RL-fBm both analytically and physically.

1. Definition and Construction

The RL-fBm of Hurst index $H$, denoted $X_H(t)$, is constructed by fractional integration of standard Brownian motion $W(t)$: $$X_H(t) = \frac{1}{\Gamma(H+\tfrac{1}{2})} \int_0^t (t - s)^{H-\frac{1}{2}} dW(s), \quad t \geq 0.$$ Here, $\Gamma$ is the Gamma function. The process is well-defined for all $H \in (0,1)$ and produces a centered Gaussian process with non-stationary increments. Notably, for $H = \frac{1}{2}$, RL-fBm reduces to standard Brownian motion $X_{1/2}(t) = W(t)$ (McGonegal, 2014, Benichou et al., 2023, Aurzada et al., 2020, Brzezniak et al., 2010).

The RL-fractional integral, $I_{0+}^{\alpha} f$ for $\alpha > 0$, is given by

$$I_{0+}^{\alpha} f(t) = \frac{1}{\Gamma(\alpha)} \int_0^t (t-s)^{\alpha-1} f(s) ds.$$

The process can equivalently be represented as the unique solution to certain fractional stochastic differential equations involving Caputo or Riemann–Liouville derivatives (Wei et al., 16 Dec 2024).

2. Covariance Structure and Sample Path Properties

The covariance function for RL-fBm is

$$\mathrm{Cov}(X_H(s), X_H(t)) = \frac{1}{\Gamma(H+\frac{1}{2})^2} \int_0^{\min(s,t)} (s-u)^{H-\frac{1}{2}} (t-u)^{H-\frac{1}{2}} du.$$

Upon evaluation via Beta and hypergeometric functions, one obtains the canonical fBm covariance

$$\mathbb{E}[X_H(t) X_H(u)] = \frac{1}{2} ( t^{2H} + u^{2H} - |t-u|^{2H} ),$$

which yields variance $\mathbb{E}[X_H(t)^2] = t^{2H}$ and expected increment scaling $$\mathbb{E}|X_H(t+\Delta) - X_H(t)|^p \leq C|\Delta|^{pH}$$, so RL-fBm admits modifications with paths Hölder continuous of any order $\gamma < H$ (McGonegal, 2014, Benichou et al., 2023, Brzezniak et al., 2010, Woszczek et al., 15 Oct 2024).

Unlike the two-sided Mandelbrot–Van Ness fBm, RL-fBm increments are not stationary except in the Brownian case. RL-fBm remains self-similar, with scaling

$$\{ X_H(at) \}_{t \geq 0} \overset{d}{=} \{ a^H X_H(t) \}_{t \geq 0}.$$

3. Path Integral, Unified Representations, and Relation to Other fBms

The Onsager–Machlup functional corresponding to RL-fBm is directly inherited from the underlying white noise: $$\mathbb{P}\{ X_H(\cdot) \} \propto \exp\left( -\frac{1}{2} \int_0^T [\dot{W}(t)]^2 dt \right).$$ In the path-integral formulation, $X_H$ is a fractional Volterra integral of white noise, and the action functional can be explicitly expressed in integral form depending on whether $H < \frac{1}{2}$ (subdiffusive) or $H > \frac{1}{2}$ (superdiffusive), differing only by integration limits or integration-by-parts regularization (Benichou et al., 2023).

A unifying representation encompasses RL-fBm (finite interval), one-sided, and two-sided (Mandelbrot–Van Ness) fBm as limit cases with different integration domains, all sharing the characteristic fractional kernel $(t-s)^{H-1/2}$ for the respective time axis. Thus, RL-fBm is canonically “FBM II” in the taxonomy of (Wei et al., 16 Dec 2024), while the MvN form yields stationary increments (Benichou et al., 2023).

4. Stochastic Calculus and Integration Theory

Classical Itô calculus is not directly applicable to RL-fBm for $H \neq \frac{1}{2}$ due to non-semimartingale character. However, integration can be achieved via:

• Pathwise Riemann–Stieltjes Integrals: If $f$ has bounded $p$-variation with $p < 1/(1-H)$, then $\int f\, d X_H$ exists pathwise (McGonegal, 2014).
• Martingale/Forward Integrals: For $H>\frac{1}{2}$, the RL-fBm admits forward stochastic integration for square-integrable adapted processes $X$, with exact $L^2$-isometry

$$\mathbb{E}\left( \int_0^T X_t\, d^{-}X_H(t) \right)^2 = \mathbb{E} \iint_{[0,T]^2} X_t X_s R_H(t,s) dt ds,$$

where $R_H(t,s)$ is the RL-fBm covariance. The integration theory employs time-dependent martingale representations and the Nelson stochastic derivative for precise operator decompositions (Costa et al., 14 Dec 2025).(Brzezniak et al., 2010) provides the Banach space generalization, with isometries and necessary $\gamma$-radonifying conditions for operator-valued integrands.

Additionally, stochastic integration rules, fractional integration by parts, and chain rules analogous to Itô’s formula are realized in the $H>\frac{1}{2}$ regime (McGonegal, 2014).

5. Non-ergodicity, Aging, and Anomalous Diffusion

Key anomalies for RL-fBm include non-ergodic time-averaged observables and spurious nonergodicity phenomena:

• Time-averaged MSD (TAMSD): For $1/2<\alpha<3/2$, the TAMSD converges to the mean squared increment (MSI), not the mean square displacement (MSD). For $\alpha \geq 3/2$, neither increments nor TAMSD are stationary or ergodic (Wei et al., 16 Dec 2024, Woszczek et al., 15 Oct 2024).
• Aging effect: With the introduction of an “aging time” $t_a$, strong aging (large $t_a$) can restore ergodicity and stationarity in the increments of RL-fBm. The scaling laws for TAMSD and MSD are altered by pre-measurement aging, with prefactor reductions but unchanged exponents (Wei et al., 16 Dec 2024).
• Diffusion exponent: The ensemble MSD scales as $t^{2\alpha-1}$ for RL-fBm generated by a fractional Langevin equation with Caputo derivative of order $\alpha > 1/2$.

A table summarizing increment and time-average behavior:

Regime	Increment Stationarity	TAMSD Limit
$1/2 < \alpha < 3/2$	Asymptotically stationary	$\to$ structure function (MSI)
$\alpha \geq 3/2$	Non-stationary	Non-stationary, $T$-dependent

6. Extensions: Random Hurst Index, Infinite Dimensions, SPDEs

Random Hurst Parameter: RL-fBm with a random Hurst exponent $\mathcal{H}$ (RL-fBMRE) models heterogeneous anomalous diffusion. Such processes are Gaussian mixtures with time-averaged quantities that mix the corresponding scaling exponents. Notably, the TAMSD and increment variance become mixtures of power laws, and ergodicity is not restored unless $\mathcal{H}$ is deterministic (Woszczek et al., 15 Oct 2024).

Infinite-Dimensional and Banach Space Extensions: Stochastic integration for RL-fBm (Liouville fBm) is developed in $L(H, E)$-valued and Hilbert space settings. For $\beta \in (0,1)$, the integrability of operator-valued processes is characterized by their membership in fractional Sobolev spaces associated with the RKHS of the driving RL-fBm. For $\beta < 1/2$, the stochastic integrals with respect to RL-fBm and classical (Mandelbrot–Van Ness type) fBm agree up to normalization (Brzezniak et al., 2010).

SPDEs: Parabolic stochastic PDEs with RL-fBm (or space-time Liouville noise) as the driving noise admit mild Hölder-continuous solutions for $\beta > d/4$ on bounded domains in $\mathbb{R}^d$ (Brzezniak et al., 2010).

7. Governing Equations, PDE Links, and Persistence Probabilities

RL-fBm arises as the scaling limit of solutions to time-dependent diffusion equations where the diffusion coefficient $D(t)$ itself solves a nonlinear Riemann–Liouville fractional ODE or integral equation:

• For $H < 1/2$ (anti-persistent), $D_t^{1-2H} D(t) = k[D(t)]^2$;
• For $H > 1/2$ (persistent), $J^{2H-1} D(t) = k[D(t)]^2$; with $D(t) = 2H Ct^{2H-1}$ yielding the exact fBm scaling (Garra et al., 2018).

Persistence exponents: The probability that RL-fBm remains below zero on $[0,T]$ decays as $\exp(-\theta^R(H) \log T )$, with $\theta^R(H) \to \infty$ as $H \to 0$ (faster than $1/H$ but slower than $1/H^2$) (Aurzada et al., 2020).

• (McGonegal, 2014): "Fractional Brownian Motion and the Fractional Stochastic Calculus"
• (Benichou et al., 2023): "A unifying representation of path integrals for fractional Brownian motions"
• (Costa et al., 14 Dec 2025): "Forward stochastic integration for adapted processes w.r.t. Riemann-Liouville fractional Brownian motion (Full version)"
• (Wei et al., 16 Dec 2024): "Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging"
• (Woszczek et al., 15 Oct 2024): "Riemann-Liouville fractional Brownian motion with random Hurst exponent"
• (Aurzada et al., 2020): "Asymptotics of the persistence exponent of integrated fractional Brownian motion and fractionally integrated Brownian motion"
• (Brzezniak et al., 2010): "Stochastic evolution equations driven by Liouville fractional Brownian motion"
• (Garra et al., 2018): "Fractional Brownian motions ruled by nonlinear equations"