Censored Fractional Bernstein Derivatives

• Censored fractional Bernstein derivatives are a generalization of fractional derivatives using Bernstein functions to suppress negative excursions and model finite-lifetime dynamics.
• They serve as the infinitesimal generators of censored Feller processes, yielding resolvent series solutions and enabling analysis of renewal and sub-diffusive behaviors.
• The framework extends classical Caputo and Marchaud operators with applications in anomalous transport, viscoelasticity, and non-local PDEs.

Censored fractional Bernstein derivatives are generalizations of fractional derivatives constructed via Bernstein functions and defined on the positive half-line. They are closely linked to censored stable subordinators and serve as the infinitesimal generators of certain Feller processes. These derivatives model dynamics where negative excursions are suppressed or "censored," resulting in processes with finite lifespan and distinct relaxation and analytic properties compared to classical (Caputo or Riemann–Liouville) fractional models. The construction, properties, and applications of censored fractional Bernstein derivatives constitute a unified theory encompassing the probabilistic, analytical, and operator-theoretic aspects associated with anomalous transport, sub-diffusion, and renewal processes.

1. Definition of the Censored Fractional Bernstein Derivative

Let $f(\lambda)$ be a complete Bernstein function (CBF) with Lévy–Khintchine representation:

$$f(\lambda) = b\lambda + \int_0^\infty (1-e^{-\lambda s})\,\mu(ds),$$

where $b\geq 0$ and $\mu$ is the Lévy measure ($\mu$ may admit a density $m$ with $m\in \mathrm{CM}$ and $\int_0^\infty (1\wedge s)\,\mu(ds)<\infty$), and let $\bar\mu(x) := \mu(x,\infty)$ denote its tail. The killing extension of a function $\phi$ is defined by $\phi^0(x) := \phi(x)$ for $x\geq0$, $0$ for $x<0$.

The Bernstein–Riemann–Liouville (BRL) fractional derivative is

$${}_0^x D^f_{\mathrm{RL}}\phi(x) = b\,\phi'(x) + \frac{d}{dx}\int_0^x \phi(t)\,\bar\mu(x-t)\,dt,$$

or in the Marchaud form,

$$D_f \phi(x) = \int_0^\infty (\phi(x)-\phi(x-s))\,\mu(ds).$$

The censored derivative modifies this by restricting the jump measure:

$${}_0^x D^f_{\mathrm{cens}} \phi(x) = \int_0^x (\phi(x) - \phi(x - s))\,\mu(ds) = D_f\phi(x) - \phi(x)\,\bar\mu(x).$$

For $f(\lambda) = \lambda^\alpha$, $\alpha \in (0,1)$, this recovers the classical censored fractional (Marchaud-type) derivative:

$$D_+^{\alpha, c} \phi(x) = \frac{\alpha}{\Gamma(1-\alpha)} \int_0^x \frac{\phi(x)-\phi(x-s)}{s^{1+\alpha}}\,ds.$$

No boundary condition is imposed at $x=0$ aside from continuity; constants lie in the kernel.

2. Generator Properties and Feller Semigroup Structure

Censored fractional Bernstein derivatives act as infinitesimal generators for censored (resurrected) decreasing subordinators. Let $S_t$ be a subordinator with Laplace exponent $f$, and define the censored process $S^c$ on $(0,\infty)$ via the Ikeda–Nagasawa–Watanabe piecing-out construction:

• Start at $x>0$.
• Run $X^1_t = x - S^1_t$ until $\sigma_1 = \inf \{ t > 0 : X^1_t < 0 \}$.
• If the process would exit at $0$, suppress that jump and restart from the pre-jump level with a new independent copy.
• Concatenate to form

$$S^c_t = \begin{cases} x - S^1_t, & 0 \leq t < \tau_1,\ S^c_{\tau_{n-1}-} - S^n_{t - \tau_{n-1}}, & \tau_{n-1} \leq t < \tau_n, \; n \geq 2,\ \partial, & t \geq \tau_\infty, \end{cases}$$

where $\tau_n$ are the exit times.

The associated semigroup $P^c_t\phi(x) = \mathbb{E}^x[\phi(S^c_t)]$ acts on the Banach space $C_\infty(0,T]$ and is strongly continuous, positivity-preserving, and satisfies the positive maximum principle. The generator is $-D_f^c$, with domain

$$\mathrm{Dom}(D_f^c) = \{ \phi \in C_\infty(0,T]: D_f^c\phi \in C_\infty(0,T] \} .$$

For $f(\lambda) = \lambda^\beta$ (stable case), the Laplace symbol of the generator remains $\phi(\lambda) = \lambda^\beta$, preserving the full jump structure.

3. Solution Theory: Resolvent Equations and Series Representation

The resolvent equation for $\phi \in C_{\bar\mu}[0,T]$ and a fixed $\lambda \in \mathbb{R}$ is

$${}_0^x D^f_{\mathrm{cens}} \phi(x) = \lambda \phi(x), \quad \phi(0) = \phi_0.$$

This is equivalent to a Volterra integral equation, typically solved via a series expansion:

$$\phi(x) = \phi_0 \sum_{j=0}^\infty (\lambda {}_0^\bullet D^f_{\mathrm{cens}})^j 1,$$

where the power-series converges uniformly on compact intervals and coincides with the Neumann series $(\lambda - {}_0^\bullet D^f_{\mathrm{cens}})^{-1}$.

For fractional cases, this specializes to a Mittag–Leffler-type expansion. In the $\beta$-stable instance, the unique solution $u(x)$ of $D_c^\beta u(x) = g(x)$ is captured by a series of Riemann–Liouville integrals and Beta-kernel operators.

4. Probabilistic Behavior: Lifetime, Hitting-Time, and Distributions

The censored subordinator $S^c$ has almost surely finite lifetime $\tau_\infty = \sup_n \tau_n$, at which it enters the cemetery state. Several explicit distributional results follow:

• $\mathbb{E}^x[\tau_n] < \infty$ for each $n$; the pre-jump value $S^c_{\tau_n} \in (0,x)$ almost surely, with density $k_n(x, r)$ recursively defined.
• The expected lifetime is

$$\mathbb{E}^x[\tau_\infty] = \sum_{n=1}^\infty \int_0^x U(0, y) k_{n-1}(x, y) dy < \infty,$$

where $U(y)$ is the potential density.

• The Laplace transform of $\tau_\infty$ solves the censored fractional relaxation equation and admits a completely monotone power-series expansion.

Table: Lifetime Distribution and Recursions

Quantity	Definition/Recursion	Notes
$k_1(x, r)$	$\bar\mu(x-r) k(r)$	Density for first pre-jump
$k_{n+1}(x, r)$	$\int_r^x k_n(x, s)\bar\mu(s-r)k(r) ds$	Density for $n$th pre-jump
$\mathbb{E}^x[e^{-\lambda \tau_\infty}]$	$\sum_{n=0}^\infty (-\lambda)^n (D_f^c)^n 1(x)$	Laplace/Mittag–Leffler expansion

5. Analytic Properties: Monotonicity, Decay, and Sub-Diffusion Regime

The solution $u(x) = \mathbb{E}^x[e^{-\lambda \tau_\infty}]$ is a completely monotone function of $x$. In the $\beta$-stable regime:

$$u(x) = 1 + \sum_{n=1}^\infty \lambda^n x^{n\beta} / \Gamma(n\beta + 1),$$

which decays at infinity as $\mathcal{O}(x^{-1-\beta})$, exactly one order faster than the Caputo relaxation solution, $O(x^{-\beta})$. This establishes a distinct class of fractional relaxation models for censored processes.

The generator’s Laplace symbol $\phi(\lambda) = f(\lambda)$ is a complete Bernstein function and its inverse operator is a Stieltjes transform. The censored derivative structure generalizes classical fractional operators to the full Bernstein scale—encompassing all subordinators with completely monotone jump spectra.

A new time-fractional diffusion equation arises:

$$D_f^c u(t, x) = L_x u(t, x), \quad u(0, x) = u_0(x),$$

with $L_x$ elliptic in $x$ and the fundamental solution given by a subordinate semigroup corresponding to the censored process, exhibiting anomalous sub-diffusive scaling (e.g., second moment $\sim t^{\beta + 1}$ for stable cases).

6. Connections, Generalizations, and Applications

Censored fractional Bernstein derivatives subsume previous constructions—for example, Caputo and Marchaud types become special cases when $f(\lambda) = \lambda^\beta$. The approach is valid for any complete Bernstein function $f$, extending the modeling range to processes controlled by arbitrary subordinators.

Applications span anomalous diffusion, viscoelasticity, and stochastic processes exhibiting kill–renewal dynamics. The unifying analytic and probabilistic framework enables rigorous treatment of initial-value problems, semigroups, and spectral theory associated with non-local, non-Markovian kernels emerging in physical and biological systems. The link to Sonine pairs (kernel and tail relations) provides a useful technical tool for operator-theoretic and convolutional analysis.

A plausible implication is that further generalizations may resolve open questions in non-local PDEs, especially those arising in the modeling of boundary-censored phenomena in random media and statistical physics. The explicit series representations, Feller properties, and connection to Bernstein functions make censored fractional Bernstein derivatives a fundamental object in the paper of subordinated processes on unbounded domains.