Generalized Fokker–Planck Equations

• Generalized Fokker–Planck equations are extensions of the classical model that incorporate memory kernels and nonlocal time behaviors to describe anomalous dynamics.
• The evolution operator method decomposes the solution into a standard propagator and a subordination kernel, linking local dynamics with complex temporal effects.
• This approach enables analytical and probabilistic interpretation, with applications ranging from subdiffusive transport in disordered materials to biological and financial systems.

Generalized Fokker–Planck equations (GFPEs) are integro-differential or operator-generalized extensions of the classical Fokker–Planck equation, designed to model non-Markovian evolution, memory effects, anomalous diffusion, and nonlocal time behaviors observed in complex systems. The evolution operator framework provides a powerful approach for representing solutions to such equations in (1+1) dimensions, incorporating memory kernels and subordination principles to connect local dynamics with anomalous spatiotemporal propagation (Górska, 14 Jan 2025).

1. Evolution Operator Representation of Solutions

The central representation for solutions to GFPEs in this framework is given by

$$q_{sM}(x, t) = \int_{0}^{\infty} d\xi\, f_{sM}(\xi, t) \, \exp(\xi L_{FP})\, q_0(x).$$

Here,

• $q_0(x)$ is the initial condition.
• $L_{FP}$ is the standard local Fokker–Planck operator (e.g., $B\partial^2_x$ for diffusion, $B\partial^2_x - \mu \partial_x$ for drift-diffusion).
• $\exp(\xi L_{FP})$ acts as the local evolution (the “parent process” propagator for operational time $\xi$).
• $f_{sM}(\xi, t)$ is a time-dependent kernel encoding memory/nonlocality, given in Laplace space by

$$f_{sM}(\xi, t) = \mathscr{L}^{-1}\left[ \widehat{M}(s)\, \exp\left(-\xi\, s\, \widehat{M}(s)\right);\,t \right],$$

where $\widehat{M}(s)$ is the Laplace transform of the memory function $M(t)$ that modulates the (possibly non-integer) time derivative in the GFPE.

Provided $\widehat{M}(s)$ is a Stieltjes function, the kernel $f_{sM}(\xi, t)$ is positive, normalized, and infinitely divisible, ensuring the operational representation is probabilistically meaningful.

2. Subordination Principle and Anomalous Kinetics

The evolution operator approach is a generalization of the subordination principle: the observable process is the result of a Markovian (local) parent process whose internal “operational time” is randomized according to the nontrivial kernel $f_{sM}(\xi, t)$. This captures the effect of memory or waiting-time distributions on the effective temporal evolution.

For power-law memory functions, such as

$$M(t) = \frac{t^{-\alpha}}{\Gamma(1-\alpha)}\,,\quad\widehat{M}(s) = s^{\alpha-1}\,,\quad 0<\alpha<1,$$

the GFPE reduces to the fractional Fokker–Planck equation (FFPE) with Caputo derivative—recovering the classical subdiffusive case. The associated subordination kernel becomes a one-sided Lévy stable (Wright–Mainardi) distribution, mirroring the broad-tailed waiting times in physical models for anomalous diffusion.

3. Extension to Generalized Diffusion–Wave Equations

The same operational approach applies to generalized diffusion–wave equations (GDWEs), where the second-order time derivative is convoluted with a memory kernel: $$\int_{0}^{t} k(t-\xi)\, \frac{\partial^2 p(x,\xi)}{\partial \xi^2} d\xi = a^2 \frac{\partial^2 p(x, t)}{\partial x^2}.$$ Here, the solution is

$$p(x, t) = \mathcal{U}(2\beta; t) p_0(x),\qquad \mathcal{U}(2\beta; t) = \int_{0}^{\infty} d\xi\, f(\beta; \xi, t)\, \mathcal{U}(2; \xi),$$

with a kernel $f(\beta; \xi, t)$ constructed from $\widehat{k}(s)$, and $\beta$ tuning the effective order of the time derivative ($\beta \in (1/2, 1]$ for fractional diffusion–wave dynamics). The method emphasizes the compositional structure of the operational solution.

4. Probabilistic Structure and Memory Functions

Key properties of the evolution operator approach emerge when the memory Laplace symbol $\widehat{M}(s)$ is a Stieltjes function:

• The subordination kernel $f_{sM}(\xi, t)$ is a legitimate probability density in its first argument $\xi$ (for fixed $t$), satisfying normalization, non-negativity, and infinite divisibility.
• For a power-law $\widehat{M}(s) = s^{\alpha-1}$, the kernel becomes the one-sided Lévy stable density—the archetypal waiting time law for subdiffusion.

Notably, whenever the “diffusion–like” initial conditions are imposed (primary initial data nonzero and all others zero), the solutions to both GFPE and GDWE have operational/probabilistic meaning, and their propagators satisfy a semigroup property. In such a scenario, the long-time scaling (e.g., the mean squared displacement) is identical for both GFPE and GDWE if their memory kernels are related by $\widehat{M}(s)^2 = \widehat{k}(s)$, underscoring their deep mathematical correspondence.

5. Physical Interpretation and Applications

The operational decomposition of GFPEs and related equations provides:

• A clear separation between spatial propagation (via $\exp(\xi L_{FP})$) and temporal complexity (memory/subordination).
• A unified way to incorporate history-dependent effects, nonlocality, and anomalous time evolution in systems as diverse as disordered materials, biological transport, and finance.
• Analytical tractability for a broad class of memory kernels, including power-law (fractional), distributed-order, and other types relevant in non-Markovian contexts.

This approach coefficients with established probabilistic interpretations, linking the abstract evolution operator treatment to subordinated stochastic processes and solidifying its position in contemporary analysis of anomalous kinetics and generalized transport.

6. Key Formulas

Component	Formula	Description
GFPE Solution	$$q_{sM}(x, t) = \int_0^\infty d\xi\, f_{sM}(\xi, t)\, \exp(\xi L_{FP})\, q_0(x)$$	Operator form of GFPE solution
Subordination Kernel	$$f_{sM}(\xi, t) = \mathscr{L}^{-1}[ \widehat{M}(s)\exp(-\xi s\widehat{M}(s));\, t ]$$	Memory function based kernel
Power-Law Memory	$$\widehat{M}(s) = s^{\alpha-1},\quad M(t) = t^{-\alpha}/\Gamma(1-\alpha),\ 0<\alpha<1$$	Caputo FFPE case
Diffusion-Wave Equation	$$p(x, t) = \int_0^\infty d\xi\, f(\beta; \xi, t)\, \mathcal{U}(2; \xi)\, p_0(x)$$	GDWE operational solution
Mean Square Displacement	Identical for GFPE and GDWE when $\widehat{M}(s)^2 = \widehat{k}(s)$ (with “diffusion-like” IC)	Common scaling for both equations

7. Summary and Synthesis

The evolution operator method for generalized Fokker–Planck equations enables explicit, operationally transparent representations of nonlocal, memory-governed processes. Solutions are mapped as convolutions of standard propagators with memory-dependent subordination kernels, conferring both analytical flexibility and direct probabilistic interpretation. When memory functions have a power-law form, the resulting dynamics correspond to fractional kinetic equations with all their characteristic anomalous phenomena. The approach unifies operational analysis with subordination and paves the way for systematic exploration of non-Markovian dynamical systems (Górska, 14 Jan 2025).