W-Fractional Diffusion Model

• W-fractional diffusion model is a generalized framework characterized by a two-parameter fractional time operator that interpolates between classical and anomalous diffusion regimes.
• It employs a Prabhakar-type kernel to implement tunable memory effects, influencing both short-time singularities and long-time decay rates.
• Applications span energy fluctuation SPDEs, stochastic lattice systems, and generalized Langevin equations, offering insights into nonlocal memory-driven dynamics.

The W-fractional diffusion model refers to a key class of generalized diffusion equations in which the time evolution is governed by a non-classical, parameter-dependent fractional time operator—typically the so-called W-operator—admitting both Volterra and nonlocal-in-time structures. Such models interpolate between standard (integer-order) diffusion and a wide spectrum of anomalous (fractional or distributed-order) diffusive behaviors, depending on the choice of operator kernel or Laplace symbol. W-fractional diffusion equations also arise as scaling limits or effective descriptions in Hamiltonian systems with multiple conserved quantities, in models with stochastic or non-conservative noise, and as constitutive relations in generalized Langevin frameworks with nonlocal memory and colored noise.

1. Operator Structure and Fundamental Definitions

At the core of the W-fractional diffusion model is the two-parameter W-operator, a fractional time derivative with Volterra structure defined by its Laplace-domain symbol

$$w(s) = s^\alpha (1 + \lambda s^{-\alpha})^{-\beta}, \qquad 0 < \alpha < 1, \quad 0 \leq \beta \leq 1, \; \lambda >0,$$

where $s$ is the Laplace dual to time. The Caputo-type W-derivative %%%%1%%%% is given in the Laplace domain by

$$\mathcal{L}\left[{}^W D_t^{\alpha,\beta} u \right](s) = w(s) \widehat u(s) - \frac{w(s)}{s} u(0).$$

Invoking the Prabhakar transform, this operator corresponds to a temporal convolution with a generalized Prabhakar-type kernel: $$K_{\alpha,\beta}(t) = t^{\alpha-1} E_{\alpha,\alpha}^{(\beta)}(-\lambda t^\alpha),$$ where $E_{\alpha,\alpha}^{(\beta)}$ is the three-parameter Prabhakar function. This formalism includes the ordinary Caputo derivative (when $\beta=0$) and introduces a modulation parameter $\beta$ that continuously adjusts the temporal memory effect. The W-operator is not generally a Bernstein function, yielding resolvent kernels and memory structures unattainable in standard fractional calculus (Wakrim, 6 Jan 2026).

2. The W-Fractional Diffusion Equation and Mild Solution Theory

The archetypal W-fractional diffusion problem is

$${}^W D_t^{\alpha,\beta} u(x,t) = \Delta u(x,t) + f(x,t), \qquad u(x,0) = u_0(x), \; u|_{\partial\Omega}=0,$$

on a domain $\Omega \subset \mathbb{R}^d$, with $A = -\Delta$ and Dirichlet boundary conditions. The mild solution is constructed using the operational calculus of the W-resolvent family: $$u(t) = R(t) u_0 + \int_0^t R(t-\tau) f(\tau)\, d\tau,$$ where the evolution operator $R(t)$ is defined via the inverse Laplace transform contour integral

$$R(t) = \frac{1}{2\pi i} \int_{\Gamma_\theta} e^{st} (w(s)I + A)^{-1} ds.$$

Well-posedness, regularity, and smoothing are guaranteed for sectorial $A$ (e.g., $-\Delta$ on $L^2$ with Dirichlet data) by resolvent estimates that exploit the distinct asymptotics of $w(s)$ for large and small $|s|$ (Wakrim, 6 Jan 2026). Each eigenmode evolves with a relaxation function $E_{\alpha,1}^{(\beta)}(-\lambda_n t^\alpha)$, generalizing the Mittag-Leffler relaxation of classical fractional models.

3. Interpolation between Standard and Anomalous Diffusion

The W-fractional diffusion model recovers a continuum between classical diffusion (exponential/Laplacian decay), standard Caputo-fractional diffusion (Mittag-Leffler temporal decay), and further regularized anomalous regimes as $\beta$ is increased. Specifically:

• For $\beta=0$, ${}^W D_t^{\alpha,\beta}$ reduces to the Caputo fractional derivative.
• For $0<\beta \leq 1$, increasing $\beta$ causes a faster algebraic decay in large-time asymptotics: $$E_{\alpha,1}^{(\beta)}(-\lambda_n t^\alpha) \sim (\lambda_n)^{-\beta} t^{-\alpha\beta}$$ as $t\to\infty$.
• The kernel $w(s)$ produces a crossover in the frequency response: Caputo-type at high frequencies, and enhanced decay at low frequencies controlled by $\beta$.

This regularization preserves short-time singularities characteristic of pure-fractional models while modifying long-time decay, effective for describing crossover phenomena in transport, population dynamics, or relaxation in complex or dispersive media (Wakrim, 6 Jan 2026).

4. Connections to Stochastic, Physical, and Lattice Models

The W-fractional framework enables precise construction of diffusive limits in out-of-equilibrium and stochastic lattice systems:

• In the Hamiltonian lattice field model with energy and volume conservation, critical scaling of a selective stochastic noise produces a limiting energy fluctuation SPDE with generator $\mathcal{L}_a$ whose symbol $Z_a(k)$ interpolates between that of Brownian motion and maximally skewed $3/2$-stable Lévy processes. The resulting model captures the transition from ordinary to anomalous superdiffusion, with the crossover parameter $a$ reflecting the energy-only noise intensity (Bernardin et al., 2016).
• In open quantum systems or Langevin models, Weyl fractional derivatives appear in the W-fractional Langevin equation, resulting in memory kernels and mean-squared displacement (MSD) scalings that interpolate between normal, subdiffusive, and superdiffusive behaviors determined by the friction order $\alpha_L=2\alpha+1$ (Vertessen et al., 2023).

These constructions demonstrate the flexibility of W-fractional calculus to encode both deterministic (through memory kernels and Prabhakar functions) and stochastic (via Lévy generator symbols and noise colorings) anomalous transport mechanisms.

5. Spectral Decomposition and Mode Relaxation

The solution admits a spectral representation: $$u(x,t) = \sum_{n=1}^\infty (u_0,\varphi_n) E_{\alpha,1}^{(\beta)}(-\lambda_n t^\alpha) \varphi_n(x) + \sum_{n=1}^\infty \int_0^t (t-\tau)^{\alpha-1} E_{\alpha,\alpha}^{(\beta)}(-\lambda_n (t-\tau)^\alpha) f_n(\tau)\, d\tau\, \varphi_n(x).$$ The modulation parameter $\beta$ determines the initial decay rate and asymptotic relaxation of each eigenmode, enabling tunable transition from slow (fractional) to fast (regularized) memory loss. For small $t$, $E_{\alpha,1}^{(\beta)}(-\lambda_n t^\alpha)$ exhibits an initial departure $\sim 1 - \beta \lambda_n t^\alpha/\Gamma(\alpha+1)$, while for large $t$, modes decay as $t^{-\alpha\beta}$ (Wakrim, 6 Jan 2026).

6. Mathematical Features: Well-Posedness, Smoothing, and Comparisons

The W-model possesses smoothing and regularity properties analogous to classical fractional diffusion:

• For any $u_0\in L^2(\Omega)$ and $f\in L^1_{\mathrm{loc}}(0,T;L^2(\Omega))$, the solution $u(t)$ is continuous on $(0,T]$ and converges strongly to $u_0$ as $t\downarrow 0$.
• For $f$ smooth, the solution is locally Hölder continuous in time.
• Smoothing rates of the evolution operator $R(t)$ are given by $\|A^\gamma R(t)\| \leq C t^{-\alpha\gamma}$ for $\gamma\in [0,1]$. These properties extend the Caputo model's behavior, with further enhancement at long times due to the regularized memory effect induced by $\beta$ (Wakrim, 6 Jan 2026).

7. Physical and Mathematical Implications

The W-fractional diffusion model provides a versatile analytic tool for capturing a wide variety of memory-driven diffusive processes. Its two-parameter kernel structure enables precise modeling of crossovers between regular and anomalous transport regimes. The non-Bernstein nature of the symbol leads to resolvent families and time-evolution operators with decay and smoothing profiles unattainable by classical or standard fractional approaches. This framework is especially suitable for processes exhibiting multi-scaling, regularized or "tempered" memory, or where interpolation between different universality classes is physically mandated.

References:

• "The W-Operator: A Volterra Fractional Time Operator with Non-Bernstein Symbol" (Wakrim, 6 Jan 2026)
• "Interpolation process between standard diffusion and fractional diffusion" (Bernardin et al., 2016)
• "Dissipative systems fractionally coupled to a bath" (Vertessen et al., 2023)