The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol

TL;DR

This work introduces a two-parameter Volterra-type fractional time operator ${}^{W}D_{t}^{\alpha,\beta}$ with symbol $\Phi_{\alpha,\beta}(s)$ that preserves Caputo-type high-frequency scaling while enabling low-frequency regularization. It shows that the natural factorization fails to yield a Bernstein function for $\beta>0$ (and is not Bernstein for $\beta>1$), placing the operator outside classical subordination. Nevertheless, the authors construct explicit Volterra representations with memory kernels $w_{\alpha,\beta}$ expressed via Prabhakar Mittag-Leffler functions and establish a fractional fundamental theorem of calculus along with a resolvent-based well-posedness theory, yielding a $W$-resolvent family. They couple this abstract theory to a $W$-fractional diffusion model to demonstrate how the modulation parameter $\beta$ tunes spectral relaxation, offering a flexible interpolation between fractional and classical diffusion while maintaining a robust operator-theoretic foundation. The results pave the way for numerical schemes based on convolution quadrature and motivate further work on spectral analysis, nonlinear extensions, and systematic numerics for the proposed model.

Abstract

We introduce a new two-parameter fractional time operator with Volterra structure, denoted by the W-operator, defined through a generalized Laplace symbol. The operator preserves the Caputo-type high-frequency behavior while allowing a controlled modification of the low-frequency regime through an additional parameter, leading to regularized memory effects. We develop a complete symbolic and Volterra theory, including explicit Prabhakar-type kernels, a left-inverse Volterra integral, and a fractional fundamental theorem of calculus. We show that the natural factorization of the Laplace symbol does not fit the classical Bernstein product mechanism and that the symbol is not a Bernstein function in general. Despite this non-Bernstein character, we establish well-posedness of abstract fractional Cauchy problems with sectorial generators by resolvent estimates and Laplace inversion, yielding a W-resolvent family with temporal regularity and smoothing properties. As an illustration, we apply the theory to a W-fractional diffusion model and discuss the influence of the modulation parameter on the relaxation of spectral modes.

Figures (3)

• Figure 1: Auxiliary function $h_{\alpha}(s)=1/(1+(1-\alpha)s^{\alpha-1})$ for $\alpha\in\{0.3,0.6,0.9\}$. The monotone increase confirms Proposition \ref{['prop:h_not_cm']}.
• Figure 2: Qualitative comparison of the kernels $k_{\mathrm{C},\alpha}(t)$ (Caputo), $k_{\mathrm{AB},\alpha}(t)$ (Atangana--Baleanu), and $w_{\alpha,\beta}(t)$ ($W$-operator) for a fixed $\alpha\in(0,1)$ and representative values of $\beta$. The Caputo kernel is singular at $t=0$, while AB kernels are bounded at the origin but typically exhibit saturation effects. The $W$-kernel preserves a transparent Volterra/Laplace structure and allows controlled modulation of the memory profile through $\beta$.
• Figure 3: Sensitivity analysis: Combined effect of $\alpha$ and $\beta$ on the energy relaxation.

Theorems & Definitions (35)

• Lemma 2.1: Regularity and asymptotics
• proof
• Proposition 2.2: Two-regime sectorial bounds
• proof
• Proposition 3.1: Non-complete monotonicity of $h_\alpha$
• proof
• Corollary 3.2: Failure of the canonical Bernstein mechanism
• Remark 3.3: On Bernstein-compatible modifications
• Proposition 3.4: Non-Bernstein property for $\beta>1$
• proof