---
title: 'Fractional Bessel Process: Approaches & Theory'
url: https://www.emergentmind.com/topics/fractional-bessel-process
type: topic
---

# Fractional Bessel Process: Approaches & Theory

Searching arXiv for recent and foundational papers on fractional Bessel processes and related Bessel-fractional operators.
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Fractional Bessel process denotes a family of non-equivalent stochastic objects obtained by fractionalizing classical Bessel dynamics in different ways. In the literature, the term has been used for the \(\gamma=0\) special case of fractional Riesz–Bessel motion, hence a stationary Gaussian process with Bessel covariance [1406.7500]; for processes generated by fractional powers of Bessel operators, equivalently stable subordinations of Bessel diffusions [1307.5019]; for Bessel-type diffusions driven by fractional Brownian motion in the rough regime \(H<1/2\) [2304.06328, 2305.15205]; and for inverse-stable time changes of Bessel diffusions with constant drift [2507.04861]. The subject therefore sits at the intersection of spectral theory, fractional calculus, Gaussian process theory, and non-Markovian diffusion.

## 1. Terminology and classical background

The classical Bessel process is the radial diffusion associated with the radial Laplacian. One common parameterization writes its generator as
\[
\mathcal{L}_\delta f(r)=\frac12 f''(r)+\frac{\delta-1}{2r}f'(r),
\]
while closely related operator-theoretic papers use
\[
B_\nu = D^2+\frac{\nu}{x}D,\qquad
B_\nu = D^2+\frac{2\nu+1}{x}D,\qquad
\Delta_\lambda=\frac{d^2}{dx^2}+\frac{2\lambda}{x}\frac{d}{dx},
\]
or
\[
B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},
\]
depending on normalization [1703.02232, 1706.01928, 1307.5019, 2006.01829]. These parameterizations are equivalent up to reindexing of the dimension or Bessel order.

A persistent source of ambiguity is that “fractional Bessel process” does not refer to a single canonical construction. In one line of work, it is a stationary Gaussian process with Bessel-function covariance; in another, it is a Markov or non-Markov process obtained from a fractional power of a Bessel generator; in another, it is an fBm-driven singular diffusion; and in another, it is a classical Bessel diffusion observed on an inverse-stable random clock [1406.7500, 2507.04861]. A useful organizing principle is to separate these constructions into: fractionalization of covariance structure, fractionalization of the spatial generator, and fractionalization of time.

## 2. Gaussian “fractional Bessel process” as a Riesz–Bessel specialization

In the survey of fractional and multifractional Gaussian processes, the main Bessel-type Gaussian field is the fractional Riesz–Bessel motion \(V_{\alpha,\gamma}\), defined in one dimension as the solution of
\[
D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t),
\qquad \alpha\ge 0,\quad 0\le \gamma<1,
\]
with \(D_t^{\gamma/2}\) the Riesz fractional derivative and \(\eta\) white noise [1406.7500]. Its spectral density is
\[
S(k)=\frac{1}{(2\pi)|k|^{2\gamma}(\omega^2+k^2)^\alpha}.
\]

In that framework, the fractional Bessel process is not the full two-parameter \(V_{\alpha,\gamma}\), but precisely the special case \(\gamma=0\). The spectrum then becomes
\[
S(k)\big|_{\gamma=0}=\frac{1}{(2\pi)(\omega^2+k^2)^\alpha},
\]
and the covariance reduces to
\[
C_{\alpha,0}(x)=\frac{2^{1/2-\alpha}}{\sqrt{\pi}\,\Gamma(\alpha)}
\left(\frac{|x|}{\omega}\right)^{\alpha-1/2}
K_{\alpha-1/2}(\omega|x|),
\]
where \(K_\nu\) is the modified Bessel function of the second kind [1406.7500]. The terminology “fractional Bessel process” here is literal: the covariance is explicitly of Bessel type.

This process is zero-mean Gaussian, stationary, and short-range dependent because its spectral density is finite at the origin. By contrast, the full fractional Riesz–Bessel motion with \(0<\gamma<1/2\) is long-range dependent, since \(S(k)\sim |k|^{-2\gamma}\) near \(k=0\) [1406.7500]. The same paper emphasizes that the two parameters separate roles that are fused in fractional Brownian motion: \(\alpha\) governs the Bessel or OU-like smoothing scale, whereas \(\gamma\) governs the Riesz singularity and memory.

The Gaussian construction also has a local regularity interpretation. For constant parameters, the effective local Hurst-type index is
\[
H=\alpha+\gamma-\frac12,
\]
so in the fractional Bessel case \(\gamma=0\), one obtains \(H=\alpha-\tfrac12\) and graph dimension \(D_H=\tfrac52-\alpha\) [1406.7500]. This makes the process locally FBM-like but globally stationary, which distinguishes it sharply from ordinary fractional Brownian motion.

## 3. Fractional powers of Bessel operators and subordinated Bessel processes

A second major meaning of fractional Bessel process comes from operator theory. Here the basic object is a fractional power of a Bessel operator on \((0,\infty)\), typically realized spectrally by the Hankel transform. For the operator
\[
\Delta_\lambda=\frac{d^2}{dx^2}+\frac{2\lambda}{x}\frac{d}{dx},
\]
the fractional power is defined by
\[
\Delta_\lambda^\sigma u = h_\lambda\!\left(x^{2\sigma}h_\lambda u\right),
\qquad 0<\sigma<1,
\]
and also admits the Balakrishnan-type semigroup formula
\[
\Delta_\lambda^\sigma u(x)
=
\frac{1}{\Gamma(-\sigma)}
\int_0^\infty \big(W_t^\lambda u(x)-u(x)\big)t^{-1-\sigma}\,dt
\]
in terms of the Bessel heat semigroup \(W_t^\lambda\) [1307.5019]. This identifies \(\Delta_\lambda^\sigma\) as the generator of a stable subordination of the Bessel diffusion, which is one of the cleanest probabilistic realizations of a fractional Bessel process.

The operator literature develops several complementary realizations of \(B_\nu^{\pm\alpha}\). One line defines negative powers as fractional Bessel integrals with explicit hypergeometric kernels, for example
\[
(IB_{\nu,-}^\alpha f)(x)
=
\frac{1}{\Gamma(2\alpha)}
\int_x^\infty
\left(\frac{y^2-x^2}{2y}\right)^{2\alpha-1}
{}_2F_1(\cdots)\,f(y)\,dy,
\]
and positive powers via composition with integer powers of the Bessel differential operator [1706.01928, 1703.02232]. These papers derive Mellin-transform formulas, group or index laws such as
\[
IB_{\nu,-}^\alpha IB_{\nu,-}^\beta = IB_{\nu,-}^{\alpha+\beta},
\]
and resolvent kernels involving Legendre, Wright, or generalized Mittag–Leffler functions [1706.01928, 1703.02232]. The same material is recast numerically through generalized Hankel translation and hypersingular-integral representations in a later computational study [2008.08082].

A Gerasimov–Caputo analogue is obtained by combining fractional Bessel integrals with integer powers of
\[
B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},
\]
namely
\[
(\mathcal{B}_{\gamma,0+}^\alpha f)(x)
=
(IB_{\gamma,0+}^{\,n-\alpha} B_\gamma^n f)(x),
\]
with Meijer-transform diagonalization
\[
\mathcal{K}_\gamma[\mathcal{B}_{\gamma,0+}^{\alpha} f](\xi)
=
\xi^{2\alpha}\mathcal{K}_\gamma[f](\xi)
\]
after suitable boundary terms vanish [2006.01829]. This provides a direct fractional-power calculus in Bessel geometry.

Across these works, the stochastic interpretation is consistent: a fractional Bessel process in the generator sense is a process on \((0,\infty)\) whose infinitesimal dynamics are governed by a fractional power of a Bessel operator. In the semigroup setting of \(\Delta_\lambda^\sigma\), the process is the stable subordination of the Bessel diffusion [1307.5019]. In the Mellin/Hankel and hypergeometric-kernel setting, the cited papers supply the resolvent, Green-kernel, and functional-calculus infrastructure needed for an explicit potential theory of such processes [1703.02232, 1706.01928, 2006.01829, 2008.08082].

## 4. Rough and fractional-diffusion Bessel processes driven by fractional Brownian motion

A distinct literature replaces the Brownian driver in a Bessel SDE by fractional Brownian motion. One starting point is the “norm-of-fBm” process
\[
\rho^H(t)=\sqrt{(B_1^H(t))^2+\cdots+(B_k^H(t))^2},
\]
which had already appeared in work of Essaky, Nualart, Guerra, and Hu, but later papers emphasize a different construction that more directly parallels the classical singular-drift Bessel SDE [2305.15205].

For \(H\in(0,1/2)\), a fractional diffusion Bessel process is introduced as the limit of regularized equations
\[
X_t^\varepsilon
=
X_0
+
a\int_0^t \frac{1}{X_s^\varepsilon\mathbf{1}_{\{X_s^\varepsilon>0\}}+\varepsilon}\,ds
+
\sigma B_t^H,
\]
and is characterized by the reflected integral equation
\[
X_t^H
=
X_0
+
a\int_0^t\frac{ds}{X_s^H}
+
\sigma B_t^H
+
L_t^H,
\]
where \(L^H\) is nondecreasing and can increase only when \(X_t^H=0\) [2304.06328]. This process is nonnegative, strictly positive for Lebesgue-a.e. time, continuous a.e. in time, and on intervals where it stays positive it satisfies the unreflected equation. Its large-time sample-path behavior is Bessel-like in scale: it eventually stays strictly positive, it dominates every \(t^\alpha\) with \(\alpha<1/2\) infinitely often, and it is eventually dominated by every \(t^\alpha\) with \(\alpha>1/2\) [2304.06328].

A closely related rough-Bessel model studies the same \(H<1/2\) regime and proves that the limiting process \(X^H\) and the associated monotone drift functional \(L^H\) are continuous [2305.15205]. In that formulation,
\[
X^H(t)=x_0+aL^H(t)+\sigma B^H(t),
\]
and, after isolating the absolutely continuous part of \(L^H\),
\[
X^H(t)
=
x_0
+
a\int_0^t \frac{1}{X^H(s)}\,ds
+
\sigma B^H(t)
+
aR^H(t),
\]
where \(R^H\) is locally constant when \(X^H>0\) and eventually constant [2305.15205]. The same paper establishes eventual strict positivity and develops consistent estimators for the Hurst index, volatility coefficient, and drift parameter. In particular, first- and second-order variations obey
\[
\left(\frac{n}{T}\right)^{-1+2H}V^n_{1,2}(X^H)\xrightarrow{\mathbb{P}}\sigma^2T,
\qquad
\left(\frac{n}{T}\right)^{-1+2H}V^n_{2,2}(X^H)\xrightarrow{\mathbb{P}}(4-2^{2H})\sigma^2T,
\]
which yields a consistent estimator of \(H\), and the long-time ratio
\[
\widehat a(T)=\frac{X^H(T)}{\int_0^T X^H(t)^{-1}\,dt}
\]
is strongly consistent for \(a\) [2305.15205].

These fBm-driven models are neither stationary Gaussian processes nor Markov processes obtained by stable subordination. Their fractional character is entirely different: it comes from rough Gaussian forcing and from the singular Bessel drift. This is one reason the term “fractional Bessel process” remains non-uniform across the literature.

## 5. Inverse-stable time change and the fractional Bessel process with constant drift

A third major construction starts from Linetsky’s Bessel diffusion with constant drift
\[
dX(t)=\left(\frac{\nu+\tfrac12}{X(t)}+\mu\right)dt+dB(t),
\qquad
\nu\in(-1,\infty)\setminus\{-1/2\},
\]
with generator
\[
\mathcal{G}f(x)
=
\left(\frac{\nu+\tfrac12}{x}+\mu\right)f'(x)+\frac12 f''(x),
\]
and then composes it with the inverse \(E(t)\) of a standard \(\alpha\)-stable subordinator \(D(t)\), \(0<\alpha<1\) [2507.04861]. The fractional Bessel process with constant drift is
\[
X_\alpha(t)=X(E(t)).
\]

The time change replaces the classical exponential spectral factors \(e^{-\lambda t}\) by Mittag–Leffler factors
\[
\mathcal{E}_\alpha(-\lambda t^\alpha)=\sum_{j=0}^\infty \frac{(-\lambda t^\alpha)^j}{\Gamma(1+\alpha j)},
\]
while preserving the spatial spectral decomposition [2507.04861]. As a result, the transition density admits a full discrete-plus-continuous spectral representation in which the discrete component is built from generalized Laguerre polynomials and the continuous component from Whittaker functions \(M_{\chi,\nu}\) and \(W_{\chi,\nu}\) [2507.04861].

The same process solves the Caputo time-fractional Cauchy problem
\[
\frac{\partial^\alpha u(y,t)}{\partial t^\alpha}
=
\mathcal{G}u(y,t),
\qquad
u(y,0)=f(y),
\]
and
\[
u_\alpha(t;y)
=
\mathbb{E}[f(X_\alpha(t))\mid X_\alpha(0)=y]
=
\int_0^\infty f(x)\,p_\alpha(x,t;y)\,dx
\]
is its strong solution [2507.04861]. This is the most direct time-fractional analogue of the Kolmogorov backward equation for a Bessel diffusion.

When \(\mu<0\), the base diffusion is positive recurrent with Gamma stationary density
\[
\pi(x)
=
\frac{-2\mu}{\Gamma(2\nu+2)}
(-2\mu x)^{2\nu+1}e^{2\mu x},
\qquad x\ge 0,
\]
and the fractional time change leaves this invariant law unchanged [2507.04861]. What changes is the rate of relaxation: the classical process converges exponentially fast, whereas the fractional process relaxes polynomially because \(\mathcal{E}_\alpha(-\lambda t^\alpha)\sim (\lambda t^\alpha\Gamma(1-\alpha))^{-1}\).

The correlation structure changes equally sharply. In stationarity, the non-fractional Bessel process has exponentially decaying correlations, whereas for the fractional process
\[
\mathrm{Corr}(X_\alpha(t),X_\alpha(s))
\sim
\frac{C(s)}{t^\alpha},
\qquad t\to\infty,
\]
for an explicit \(C(s)\), so the process exhibits long-range dependence [2507.04861]. This paper also uses the fractional model as a heavy-traffic limit for a polling system with random malfunction periods in queueing theory, where the inverse stable subordinator models inactive server episodes [2507.04861].

## 6. Special-function, series, and numerical frameworks

Deterministic fractional Bessel equations supply much of the special-function backbone for the process theory. In the conformable setting, the sequential conformable fractional Bessel equation
\[
x^{2\alpha} T_\alpha T_\alpha y+\alpha x^\alpha T_\alpha y+\alpha^2(x^{2\alpha}-\nu^2)y=0
\]
reduces to the classical Bessel equation when \(\alpha=1\), and its fractional Bessel functions are essentially classical Bessel functions evaluated at \(x^\alpha\); for example,
\[
(J_\alpha)_p(x)
=
\sum_{n=0}^\infty
\frac{(-1)^n}{n!\,\Gamma(p+n+1)}
\left(\frac{x^\alpha}{2}\right)^{2n+p}
\]
and
\[
(J_\alpha)_{1/2}(x)=\sqrt{\frac{2}{\pi x^\alpha}}\sin(x^\alpha)
\]
[1506.07382]. This does not define a stochastic process, but it furnishes an explicit eigenfunction calculus for conformable Bessel-type operators.

A broader multi-term fractional Bessel equation,
\[
\sum_{i=1}^{m}d_i\,x^{\alpha_i}D^{\alpha_i}u(x)+(x^\beta-\nu^2)u(x)=0,
\]
admits fractional or logarithmic fractional power-series solutions determined by the characteristic equation
\[
\sum_{i=1}^{m}d_i\frac{\Gamma(1+\gamma)}{\Gamma(1+\gamma-\alpha_i)}=\nu^2,
\]
and the theory identifies when the series solution is unique, non-unique, or fails to exist [2112.13516]. A quasi-Bessel extension with shifted powers,
\[
\sum_{i=1}^{m} d_i x^{\alpha_i+p_i}D^{\alpha_i}u(x)+(x^\beta-\nu^2)u(x)=0,
\]
shows that matching the highest-order derivative with a pure Bessel power \(p_1=0\) is structurally necessary for the fractional-series method; it also yields threshold conditions on \(\nu^2\) and uniqueness in \(C^l\) by a contraction argument [2201.10094]. These results suggest a boundary-value and spectral theory for generalized fractional Bessel generators, although the stochastic process itself is not constructed there.

On the numerical and harmonic-analysis side, a recent chromatic-expansion framework starts from the Bessel–Laplace operator \(\Delta_a\) and its spectral fractional powers
\[
(\mathcal{H}_a(-\Delta_a)^{s/2}\varphi)(\xi)=|\xi|^s\hat\varphi(\xi),
\]
together with the nonlocal integral representation
\[
(-\Delta_a)^{s/2}\varphi(x)
=
C(d,a,s)\int_{\mathbb{R}_+^d}
\frac{\varphi(x)-T_a^y\varphi(x)}{|y|^{d+|a|+s}}\,y^{-a}\,dy
\]
[2606.26737]. The same paper proves the intertwining
\[
(-B_\gamma)^{k+s/2}_t\,M_{sph,a}(f)(x,t)
=
M_{sph,a}\big((-\Delta_a)^{k+s/2}_x f\big)(x,t),
\]
which reduces radial fractional Bessel dynamics to one-dimensional Bessel operators on the spherical-mean variable \(t\) [2606.26737]. This gives a concrete approximation strategy for semigroups and transition kernels of radial fractional Bessel operators.

A separate analytic development introduces pseudo-differential operators associated with a fractional Hankel–Bessel transform, together with symbol classes, kernel estimates, and weighted Sobolev spaces; no stochastic process is defined there, but this suggests a natural pseudo-differential calculus for Bessel-type Lévy generators in a fractional Hankel setting [2601.03091].

The modern literature therefore supports an important negative conclusion as well as a positive one. The negative conclusion is that there is no single object called the fractional Bessel process. The positive conclusion is that the main constructions are now reasonably well separated: stationary Gaussian Bessel-covariance models [1406.7500], spatial fractionalizations of Bessel generators [1307.5019, 1703.02232], rough fBm-driven singular diffusions [2304.06328, 2305.15205], and inverse-stable time changes of Bessel diffusions with explicit spectral theory [2507.04861]. A plausible implication is that future unification will proceed not by fixing a single definition, but by treating “fractional Bessel process” as a class name indexed by the mechanism of fractionalization.

Source: https://www.emergentmind.com/topics/fractional-bessel-process