Hadamard Fractional Brownian Motion

• Hadamard fractional Brownian motion is a Gaussian process defined via Hadamard fractional operators with logarithmic kernels, ensuring N(0,t) marginals.
• It is self-similar with a constant Hurst index of 1/2 and exhibits both short/anti-persistent and long-range memory based on the parameter α.
• The process features precise path regularity, robust stochastic integration, and extends to modeling fractional PDEs and ultra-slow diffusions in grey-noise settings.

Hadamard fractional Brownian motion (HfBm) is a class of Gaussian processes constructed via Hadamard-type fractional integral and derivative operators rather than the classical Riemann–Liouville or Weyl kernels. The central feature distinguishing HfBm from both standard Brownian motion (Bm) and classical fractional Brownian motion (fBm) is the logarithmic kernel induced by Hadamard fractional calculus. While its one-dimensional distributions coincide with those of Bm, HfBm embodies a range of long- and short-memory behaviors, generalized self-similarity (with Hurst index $H=1/2$ for all parameter values), and sharply defined path regularity. Its stochastic integration theory, inverse representations via multiplicative Sonine pairs, and law of the iterated logarithm are well-developed. Extensions to grey-noise spaces controlled by Le Roy measures further link HfBm to generalized diffusions governed by evolutionary PDEs with Hadamard-type time derivatives (Beghin et al., 17 Jul 2025, Beghin et al., 2024).

1. Construction via Hadamard Fractional Operators

HfBm is constructed using right-sided Hadamard fractional integrals and derivatives defined for $\beta>0$ as

$${}_{\mathcal H}I_{-}^{\beta}f(x) = \frac{1}{\Gamma(\beta)} \int_{x}^{\infty} \left(\ln\frac{z}{x}\right)^{\beta-1} \frac{f(z)}{z}\,dz,$$

with the fractional derivative

$${}_{\mathcal H}D_{-}^{\beta}f(x) = \left(-x \frac{d}{dx}\right)^n \left[x^n {}_{\mathcal H}I_{-}^{n-\beta}f(x)\right],\quad n = \lfloor\Re(\beta)\rfloor+1,$$

satisfying $${}_{\mathcal H}D_{-}^{\beta}\circ {}_{\mathcal H}I_{-}^{\beta} = \mathrm{Id}$$ for admissible functions. The canonical kernel operator is

$${}_{\mathcal H}M_-^\alpha f = K_\alpha \begin{cases} {}_{\mathcal H}D_{-}^{(1-\alpha)/2}f & \alpha\in(0,1), \ f & \alpha=1, \ {}_{\mathcal H}I_{-}^{(\alpha-1)/2}f & \alpha\in(1,2), \end{cases}$$

with $$K_\alpha = \frac{\Gamma((\alpha+1)/2)}{\sqrt{\Gamma(\alpha)}}$$ ensuring that the marginal distribution at time $t$ is $N(0,t)$.

Defining the process in the white noise space $(\mathcal S, \nu)$,

$$B^\mathcal H_\alpha(t, \omega) = \langle \omega, {}_{\mathcal H}M_-^\alpha 1_{[0, t)} \rangle,$$

which yields a centered Gaussian process with covariance

$\Cov(B^\mathcal H_\alpha(s), B^\mathcal H_\alpha(t)) = \frac{1}{\Gamma(\alpha)} \int_0^{s\wedge t} \left(\ln\frac{t}{u}\right)^{\frac{\alpha-1}{2}} \left(\ln\frac{s}{u}\right)^{\frac{\alpha-1}{2}}\, du.$

An equivalent Volterra-type representation holds: $$B^\mathcal H_\alpha(t) = \frac{1}{\sqrt{\Gamma(\alpha)}} \int_0^t \left(\ln\frac{t}{s}\right)^{\frac{\alpha-1}{2}} dB(s),$$ where $B(s)$ is a standard Brownian motion.

2. Self-Similarity and Memory Properties

HfBm is self-similar with index $H=1/2$ for all $\alpha$: $$B^\mathcal H_\alpha(c t) \stackrel{d}{=} c^{1/2} B^\mathcal H_\alpha(t), \quad c>0.$$ The increments exhibit distinct memory regimes dependent on $\alpha$:

• For $\alpha\in(0,1)$: $\Cov(\Delta_n, \Delta_{n+k}) \sim -k^{\alpha-2}$ with $\sum_k |\Cov| < \infty$, resulting in short or antipersistent memory.
• For $\alpha\in(1,2)$: $\Cov(\Delta_n, \Delta_{n+k}) \sim +k^{\alpha-2}$ and $\sum_k \Cov = \infty$, resulting in long-range dependence (Beghin et al., 17 Jul 2025, Beghin et al., 2024).

A comparison with classical fBm shows that for $\alpha\in(1,2)$ the divergence rate of the partial-sum variance $m^{\alpha-1}$ is strictly slower than for fBm of Hurst $H=\alpha/2$; hence, the memory effect in HfBm is “weaker” (Beghin et al., 2024).

3. Pathwise Regularity and Trajectory Properties

Increment variances satisfy sharp bounds: $V(s, t) = \E\left[(B^\mathcal H_\alpha(t) - B^\mathcal H_\alpha(s))^2\right] \le \begin{cases} C_{T, \alpha}(t-s)^\alpha, & \alpha\in(0,1), \ C_\alpha(t-s), & \alpha\in(1,2), \end{cases} \quad 0 \leq s < t \leq T.$ Consequently, sample paths are almost surely Hölder continuous: $$B^\mathcal H_\alpha \in \begin{cases} C^{\alpha/2-}([0,T]), & \alpha\in(0,1), \ C^{1/2-}([0,T]), & \alpha\in(1,2). \end{cases}$$ Generalized quasi-helix bounds are established; for $\alpha\in(0,1)$ the process is a $(1,\alpha,T)$-generalized quasi-helix, and for $\alpha \in (1,2)$ a $(\alpha,1,T)$-generalized quasi-helix.

Exact $p$-variation is determined by

$\E V^p_n \to \begin{cases} \text{finite} & p = 2/\alpha, \ 0 & p > 2/\alpha, \ \infty & p < 2/\alpha, \end{cases} \quad n \to \infty.$

For $\alpha > 1$, the quadratic variation ($p=2$) vanishes.

Local nondeterminism is verified via the Volterra representation: for any partition $0 < t_1 < \dots < t_m < T$, the conditional variance of increments is bounded below by the unconditional increment variance.

4. Stochastic Integration and Inverse Representation

The space of admissible integrands is

$$L_{2, \alpha} = \{ f: \R^+ \to \R \mid {}_{\mathcal H}M_-^\alpha f \in L^2(\R^+)\}, \quad \|f\|_{L_{2, \alpha}}^2 = \| {}_{\mathcal H}M_-^\alpha f \|_{L^2}^2,$$

with integration defined by

$$\int_0^\infty f(s) dB^\mathcal H_\alpha(s) := \int_0^\infty ({}_{\mathcal H}M_-^\alpha f)(s) dB(s).$$

This extends uniquely to an isometry $I_\alpha: L_{2, \alpha} \to L^2(\Omega)$.

For smooth integrands $f \in C^\beta([0,T])$ with $\beta + H > 1$, the Riemann–Stieltjes integral $\int_0^T f(s) dB^\mathcal H_\alpha(s)$ exists and obeys

$$\int_0^T f\, dB^\mathcal H_\alpha = f(T) B^\mathcal H_\alpha(T) - \int_0^T B^\mathcal H_\alpha(s) f'(s) ds.$$

The inverse representation utilizes the multiplicative Sonine pair property of two power-law kernels: $${}^{\mathcal H}\overline M_-^\alpha = \frac{1}{K_\alpha} \begin{cases} {}_{\mathcal H}I_-^{(1-\alpha)/2}, & \alpha \in (0,1), \ \mathrm{Id}, & \alpha = 1, \ {}_{\mathcal H}D_-^{(\alpha-1)/2}, & \alpha \in (1,2), \end{cases}$$ yielding

$$B(t) = K_\alpha' \int_0^\infty ({}^{\mathcal H}\overline M_-^\alpha 1_{[0, t)})(s) dB^\mathcal H_\alpha(s), \quad K_\alpha' = 1/K_\alpha.$$

The Mellin convolution of the logarithmic kernels involved is constant.

5. Reproducing Kernel Hilbert Space and Limit Laws

The RKHS associated with HfBm is described by functions $F = Af$ for $f \in L^2(\R^+)$,

$$(Af)(t) = \frac{1}{\sqrt{\Gamma(\alpha)}} \int_0^t \left(\ln \frac{t}{s}\right)^{\frac{\alpha-1}{2}} f(s)\, ds,$$

with inverse given via the Sonine dual integrals as above.

The law of iterated logarithm (LIL) is established, both at zero and as $t\to\infty$:

• As $u\downarrow 0$, the family $$\{B^\mathcal H_\alpha(ut)/\sqrt{2u \ln\ln(1/u)} : t \in [0,T]\}$$ is relatively compact in $C([0,T])$ with its set of limit points being the unit ball of the associated RKHS.
• As $n \to \infty$, the discrete family $$\{B^\mathcal H_\alpha(nt)/\sqrt{2n \ln\ln n} : t \in [0,T]\}$$ is relatively compact in the same topology and shares the same cluster set.

6. Extensions in Gel'fand Sense and Fractional PDEs

Generalized random processes related to Hadamard operators have been constructed in both white-noise and grey-noise spaces. In the latter, the underlying measure is induced by the Le Roy function $R_B$, providing a non-Gaussian extension (“Le Roy–Hadamard motion”). The one-dimensional distributions satisfy heat equations with non-constant coefficients and fractional Hadamard time-derivatives, specifically involving Caputo-type Hadamard fractional derivatives. This construction enables modeling of ultra-slow diffusions while preserving Gaussianity for one-dimensional marginals within any finite time horizon (Beghin et al., 2024).

In these generalized settings, distributional derivatives and stochastic differential equations (e.g., Hadamard-fractional Ornstein–Uhlenbeck processes) can be formulated. The existence and explicit form of distributional (Gel'fand) derivatives and their $S$-transforms are established: $S(N^H_a(t))(\varphi) = K_a \Gamma(1-\frac{a}{2})\, (\Hi_{0+}^{(1+a)/2}\varphi)(t),$ and the moving-average representation

$$N^H_a(t) = \int_0^t \partial_t [M^{a/2} 1_{[0,t)}](s)\, dB(s).$$

Explicit solutions for Ornstein–Uhlenbeck type equations driven by HfBm are obtained.

7. Summary Table: Key Features of Hadamard Fractional Brownian Motion

Feature	Classical Bm / fBm	HfBm (white-noise construction)
Marginal Law	$N(0,t)$ (Bm) / $N(0,\sigma^2 t^{2H})$ (fBm)	$N(0,t)$
Kernel	Power-law (R-L)	Logarithmic (Hadamard)
Hurst Index	$H=1/2$ (Bm); $H\in(0,1)$ (fBm)	$H=1/2$ for all $\alpha$
Memory (increments)	Short (Bm); long (fBm $H>1/2$)	$\alpha\in(0,1)$: short/anti-persistent; $\alpha\in(1,2)$: long-range
Path Regularity	Hölder $1/2-$ (Bm); $H-$ (fBm)	$\alpha\in(0,1)$: $\alpha/2-$; $\alpha\in(1,2)$: $1/2-$
Quadratic Variation	Non-vanishing (Bm); vanishes (fBm $H>1/2$)	Vanishes for $\alpha>1$
RKHS	$L^2([0,T])$ (Bm); Volterra-type (fBm)	Volterra-type via Hadamard kernels

The Hadamard fractional Brownian motion provides a rigorous framework for studying processes with logarithmic kernel memory, ultra-slow diffusion behavior, and complex path regularity, with foundational results for integration, limit laws, and extensions to broader noise spaces (Beghin et al., 17 Jul 2025, Beghin et al., 2024).