Fractional Bessel Process: Approaches & Theory

• Fractional Bessel processes are a family of stochastic objects derived by fractionalizing classical Bessel dynamics via covariance, spatial generators, or time changes.
• They are analyzed using diverse methods including spectral theory, fractional calculus, stable subordination, and fBm-driven diffusions to capture distinct process behaviors.
• These models exhibit features such as stationary Gaussian covariance, long-range dependence, and explicit spectral representations, enhancing advanced stochastic analysis.

Searching arXiv for recent and foundational papers on fractional Bessel processes and related Bessel-fractional operators. {"^^^^^^^^4query4 Bessel\"^^^^^^^^4 OR abs:\4^^^^^^^^"fractional Bessel process\"^^^^^^^^4 OR abs:\4^^^^^^^^"fractional Bessel\"","max_results":^^^^4ti:\4query4^^^^,"sort_by":"submittedDate","sort_order":"descending"} 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 PRESERVED_PLACEHOLDER_^^^^4query4^^^^ special case of fractional Riesz–Bessel motion, hence a stationary Gaussian process with Bessel covariance (&&&^^^^4query4^^^^&&&); for processes generated by fractional powers of Bessel operators, equivalently stable subordinations of Bessel diffusions (&&&^^^^4ti:\4^^^^&&&); for Bessel-type diffusions driven by fractional Brownian motion in the rough regime PRESERVED_PLACEHOLDER_^^^^4ti:\4^^^^ (&&&^^^^4 OR abs:\4^^^^&&&, &&&^^^^4 OR abs:\4^^^^&&&); and for inverse-stable time changes of Bessel diffusions with constant drift (Papić, 7 Jul 2025). The subject therefore sits at the intersection of spectral theory, fractional calculus, Gaussian process theory, and non-Markovian diffusion.

^^^^4ti:\4^^^^. Terminology and classical background

The classical Bessel process is the radial diffusion associated with the radial Laplacian. One common parameterization writes its generator as

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^

while closely related operator-theoretic papers use

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^

or

$$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$

depending on normalization (Shishkina et al., 2017, Sitnik et al., 2017, &&&^^^^4ti:\4^^^^&&&, Shishkina et al., 2020). 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 (&&&^^^^4query4^^^^&&&, Papić, 7 Jul 2025). A useful organizing principle is to separate these constructions into: fractionalization of covariance structure, fractionalization of the spatial generator, and fractionalization of time.

^^^^4 OR abs:\4^^^^. 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 (&&&^^^^4query4^^^^&&&). 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 PRESERVED_PLACEHOLDER_^^^^4ti:\4query4^^^^, but precisely the special case PRESERVED_PLACEHOLDER_^^^^4ti:\4ti:\4^^^^. The spectrum then becomes

PRESERVED_PLACEHOLDER_^^^^4ti:\4 OR abs:\4^^^^

and the covariance reduces to

PRESERVED_PLACEHOLDER_^^^^4ti:\4 OR abs:\4^^^^

where PRESERVED_PLACEHOLDER_^^^^4ti:\44^^^^ is the modified Bessel function of the second kind (&&&^^^^4query4^^^^&&&). 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 PRESERVED_PLACEHOLDER_^^^^4ti:\4^^^^5 is long-range dependent, since PRESERVED_PLACEHOLDER_^^^^4ti:\4^^^^6 near PRESERVED_PLACEHOLDER_^^^^4ti:\4^^^^7 (&&&^^^^4query4^^^^&&&). The same paper emphasizes that the two parameters separate roles that are fused in fractional Brownian motion: PRESERVED_PLACEHOLDER_^^^^4ti:\4^^^^8 governs the Bessel or OU-like smoothing scale, whereas PRESERVED_PLACEHOLDER_^^^^4ti:\4^^^^9 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

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^

so in the fractional Bessel case PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4ti:\4^^^^, one obtains PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^ and graph dimension PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^ (&&&^^^^4query4^^^^&&&). This makes the process locally FBM-like but globally stationary, which distinguishes it sharply from ordinary fractional Brownian motion.

^^^^4 OR abs:\4^^^^. 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 PRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^, typically realized spectrally by the Hankel transform. For the operator

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5

the fractional power is defined by

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6

and also admits the Balakrishnan-type semigroup formula

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7

in terms of the Bessel heat semigroup PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8 (&&&^^^^4ti:\4^^^^&&&). This identifies PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9 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 PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4query4^^^^. One line defines negative powers as fractional Bessel integrals with explicit hypergeometric kernels, for example

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4ti:\4^^^^

and positive powers via composition with integer powers of the Bessel differential operator (Sitnik et al., 2017, Shishkina et al., 2017). These papers derive Mellin-transform formulas, group or index laws such as

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^

and resolvent kernels involving Legendre, Wright, or generalized Mittag–Leffler functions (Sitnik et al., 2017, Shishkina et al., 2017). The same material is recast numerically through generalized Hankel translation and hypersingular-integral representations in a later computational study (&&&^^^^4 OR abs:\4query4^^^^&&&).

A Gerasimov–Caputo analogue is obtained by combining fractional Bessel integrals with integer powers of

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4 OR abs:\4^^^^

namely

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\44^^^^

with Meijer-transform diagonalization

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^5

after suitable boundary terms vanish (Shishkina et al., 2020). 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 PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^6 whose infinitesimal dynamics are governed by a fractional power of a Bessel operator. In the semigroup setting of PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^7, the process is the stable subordination of the Bessel diffusion (&&&^^^^4ti:\4^^^^&&&). 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 (Shishkina et al., 2017, Sitnik et al., 2017, Shishkina et al., 2020, &&&^^^^4 OR abs:\4query4^^^^&&&).

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

PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^8

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 (&&&^^^^4 OR abs:\4^^^^&&&).

For PRESERVED_PLACEHOLDER_^^^^4 OR abs:\4^^^^9, a fractional diffusion Bessel process is introduced as the limit of regularized equations

$$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$^^^^4query4^^^^

and is characterized by the reflected integral equation

$$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$^^^^4ti:\4^^^^

where $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$^^^^4 OR abs:\4^^^^ is nondecreasing and can increase only when $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$^^^^4 OR abs:\4^^^^ (&&&^^^^4 OR abs:\4^^^^&&&). 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 $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$4 with $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$5 infinitely often, and it is eventually dominated by every $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$6 with $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$7 (&&&^^^^4 OR abs:\4^^^^&&&).

A closely related rough-Bessel model studies the same $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$8 regime and proves that the limiting process $$B_\gamma=\frac{d^2}{dx^2}+\frac{\gamma}{x}\frac{d}{dx},$$9 and the associated monotone drift functional $V_{\alpha,\gamma}$^^^^4query4^^^^ are continuous (&&&^^^^4 OR abs:\4^^^^&&&). In that formulation,

$V_{\alpha,\gamma}$^^^^4ti:\4^^^^

and, after isolating the absolutely continuous part of $V_{\alpha,\gamma}$^^^^4 OR abs:\4^^^^,

$V_{\alpha,\gamma}$^^^^4 OR abs:\4^^^^

where $V_{\alpha,\gamma}$4 is locally constant when $V_{\alpha,\gamma}$5 and eventually constant (&&&^^^^4 OR abs:\4^^^^&&&). 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

$V_{\alpha,\gamma}$6

which yields a consistent estimator of $V_{\alpha,\gamma}$7, and the long-time ratio

$V_{\alpha,\gamma}$8

is strongly consistent for $V_{\alpha,\gamma}$9 (&&&^^^^4 OR abs:\4^^^^&&&).

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

$$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$^^^^4query4^^^^

with generator

$$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$^^^^4ti:\4^^^^

and then composes it with the inverse $$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$^^^^4 OR abs:\4^^^^ of a standard $$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$^^^^4 OR abs:\4^^^^-stable subordinator $$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$4, $$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$5 (Papić, 7 Jul 2025). The fractional Bessel process with constant drift is

$$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$6

The time change replaces the classical exponential spectral factors $$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$7 by Mittag–Leffler factors

$$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$8

while preserving the spatial spectral decomposition (Papić, 7 Jul 2025). 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 $$D_t^{\gamma/2}(D_t+\omega)^{\alpha/2}V_{\alpha,\gamma}=\eta(t), \qquad \alpha\ge 0,\quad 0\le \gamma<1,$$9 and $D_t^{\gamma/2}$^^^^4query4^^^^ (Papić, 7 Jul 2025).

The same process solves the Caputo time-fractional Cauchy problem

$D_t^{\gamma/2}$^^^^4ti:\4^^^^

and

$D_t^{\gamma/2}$^^^^4 OR abs:\4^^^^

is its strong solution (Papić, 7 Jul 2025). This is the most direct time-fractional analogue of the Kolmogorov backward equation for a Bessel diffusion.

When $D_t^{\gamma/2}$^^^^4 OR abs:\4^^^^, the base diffusion is positive recurrent with Gamma stationary density

$D_t^{\gamma/2}$4

and the fractional time change leaves this invariant law unchanged (Papić, 7 Jul 2025). What changes is the rate of relaxation: the classical process converges exponentially fast, whereas the fractional process relaxes polynomially because $D_t^{\gamma/2}$5.

The correlation structure changes equally sharply. In stationarity, the non-fractional Bessel process has exponentially decaying correlations, whereas for the fractional process

$D_t^{\gamma/2}$6

for an explicit $D_t^{\gamma/2}$7, so the process exhibits long-range dependence (Papić, 7 Jul 2025). 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 (Papić, 7 Jul 2025).

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

$D_t^{\gamma/2}$8

reduces to the classical Bessel equation when $D_t^{\gamma/2}$9, and its fractional Bessel functions are essentially classical Bessel functions evaluated at $\eta$^^^^4query4^^^^; for example,

$\eta$^^^^4ti:\4^^^^

and

$\eta$^^^^4 OR abs:\4^^^^

(&&&4^^^^4query4^^^^&&&). 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,

$\eta$^^^^4 OR abs:\4^^^^

admits fractional or logarithmic fractional power-series solutions determined by the characteristic equation

$\eta$4

and the theory identifies when the series solution is unique, non-unique, or fails to exist (&&&4^^^^4ti:\4^^^^&&&). A quasi-Bessel extension with shifted powers,

$\eta$5

shows that matching the highest-order derivative with a pure Bessel power $\eta$6 is structurally necessary for the fractional-series method; it also yields threshold conditions on $\eta$7 and uniqueness in $\eta$8 by a contraction argument (&&&4^^^^4 OR abs:\4^^^^&&&). 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 $\eta$9 and its spectral fractional powers

$$S(k)=\frac{1}{(2\pi)|k|^{2\gamma}(\omega^2+k^2)^\alpha}.$$^^^^4query4^^^^

together with the nonlocal integral representation

$$S(k)=\frac{1}{(2\pi)|k|^{2\gamma}(\omega^2+k^2)^\alpha}.$$^^^^4ti:\4^^^^

(&&&4^^^^4 OR abs:\4^^^^&&&). The same paper proves the intertwining

$$S(k)=\frac{1}{(2\pi)|k|^{2\gamma}(\omega^2+k^2)^\alpha}.$$^^^^4 OR abs:\4^^^^

which reduces radial fractional Bessel dynamics to one-dimensional Bessel operators on the spherical-mean variable $$S(k)=\frac{1}{(2\pi)|k|^{2\gamma}(\omega^2+k^2)^\alpha}.$$^^^^4 OR abs:\4^^^^ (&&&4^^^^4 OR abs:\4^^^^&&&). 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 (Pasawan, 6 Jan 2026).

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 (&&&^^^^4query4^^^^&&&), spatial fractionalizations of Bessel generators (&&&^^^^4ti:\4^^^^&&&, Shishkina et al., 2017), rough fBm-driven singular diffusions (&&&^^^^4 OR abs:\4^^^^&&&, &&&^^^^4 OR abs:\4^^^^&&&), and inverse-stable time changes of Bessel diffusions with explicit spectral theory (Papić, 7 Jul 2025). 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.