Abstract

This paper develops a complete mathematical theory for degenerate kinetic Fokker-Planck equations with multiplicative Lévy noise. The main novelty is the treatment of state-dependent noise intensity $\sigma \left(x,v\right)$ that may vanish on sets of positive measure, a situation arising naturally in applications with heterogeneous media or discontinuous coefficients. We establish three fundamental contributions: 1) a well-posedness theory for solutions in fractional Sobolev spaces under minimal regularity assumptions on $\sigma$ ; 2) a critical degeneracy threshold theorem proving that hypoelliptic regularization persists even when $\sigma \left(x,v\right)$ vanishes, provided the degeneracy measure satisfies $\delta <\alpha {\lambda }_1/\left(2+\alpha \right)$ , where ${\lambda }_1$ is the mixing rate associated with the transport operator; 3) rigorous hydrodynamic limits yielding effective fractional diffusion equations with spatially-dependent coefficients $\overline{\sigma }\left(x\right)$ . The analysis combines weighted energy methods, fractional commutator estimates, and compensated compactness techniques to handle the interplay between degeneracy and nonlocality. Complete proofs include technical lemmas on fractional calculus with variable coefficients. Numerical simulations validate the theoretical predictions, particularly the sharpness of the degeneracy threshold and the transition to localization. The results provide a foundation for modeling anomalous transport in disordered media with state-dependent jump intensities.

Keywords

Kinetic Equations, Fokker-Planck Equation, Lévy Noise, Fractional Hypoellipticity, Multiplicative Noise, Degenerate Coefficients, Hydrodynamic Limits, Fractional Calculus

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1. Introduction

Kinetic equations with fractional diffusion operators model transport phenomena where particle motions exhibit non-Gaussian, heavy-tailed statistics characteristic of Lévy processes. Such models arise in plasma physics [1], anomalous diffusion in disordered media [2], and financial markets [3]. A fundamental mathematical challenge emerges when the noise intensity depends on the state variables $\left(x,v\right)$ , particularly when it may vanish or become arbitrarily small on significant portions of the phase space. This degenerate multiplicative noise scenario occurs naturally in heterogeneous media, boundary layers, or systems with discontinuous transport coefficients.

While the constant-coefficient case is well understood [4], the treatment of variable coefficients in fractional kinetic equations remains largely unexplored. The present work bridges this gap by developing a comprehensive theory for the degenerate kinetic Fokker-Planck equation with multiplicative Lévy noise:

${\partial }_{t}f+v\cdot {\nabla }_{x}f=-\sigma \left(x,v\right){\left(-{\text{Δ}}_v\right)}^{\alpha /2}f+{\nabla }_v\cdot \left(Ff\right),$ (1)

where $0<\alpha <2$ and $\sigma :{ℝ}^{2d}\to \left[0,\infty \right)$ may vanish on sets of positive measure.

1.1. State of the Art

The mathematical literature on kinetic equations can be divided into three strands relevant to our work:

1) Constant-coefficient fractional kinetics: Hypocoercivity and hydrodynamic limits for equations with ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}$ are established in [4] [5]. These works assume $\sigma \equiv 1$ and rely on Fourier analysis techniques that break down for variable coefficients.

2) Multiplicative Gaussian noise: For $\alpha =2$ , degenerate parabolic equations with variable coefficients have been studied via Hörmander’s theory [6] and its hypoelliptic extensions. Recent advances in hypocoercivity for degenerate kinetic equations appear in [7], but these concern local diffusion ($\alpha =2$ ).

3) Fractional operators with variable coefficients: In elliptic/parabolic settings, the Kato-Ponce commutator estimates [8] and recent advances in nonlocal calculus [9] provide tools. However, their application to kinetic equations with degenerate coefficients is novel.

1.2. Recent Developments (Post-2020)

Recent advances in fractional kinetic equations with variable coefficients include:

• Time-dependent coefficients: The work of [10] establishes well-posedness for kinetic equations with time-dependent $\sigma \left(t,x,v\right)$ using evolution semigroup methods, but requires uniform ellipticity ($\sigma \ge {\sigma }_{\text{min}}>0$ ).

• Anisotropic fractional diffusion: [11] considers anisotropic fractional operators ${\displaystyle {\sum }_{i=1}^d{\sigma }_i\left(x,v\right){\left(-{\partial }_{v_i}^2\right)}^{{\alpha }_i/2}}$ but assumes ${\sigma }_i\in C^{\infty }$ with no degeneracy.

• Nonlinear fractional kinetics: [12] studies nonlinear collision operators coupled with fractional diffusion, yet treats $\sigma$ as constant.

• Numerical methods: Recent schemes by [13] for fractional Fokker-Planck equations with variable coefficients use spectral methods but assume $\sigma \left(x,v\right)$ is strictly positive and smooth.

Our work differs fundamentally by: 1) allowing $\sigma$ to vanish on sets of positive measure (controlled degeneracy), 2) establishing a sharp degeneracy threshold for hypoellipticity, and 3) handling the combined challenges of nonlocality, degeneracy, and kinetic transport simultaneously. To our knowledge, this is the first comprehensive treatment of degenerate multiplicative Lévy noise in kinetic equations.

The principal difficulty in analyzing (1) lies in the interaction between three features: 1) the nonlocal nature of ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}$ ; 2) the degeneracy of $\sigma \left(x,v\right)$ ; 3) the coupling between $x$ and $v$ via transport $v\cdot {\nabla }_x$ . Classical methods from each separate literature are insufficient for this combination.

1.3. Novelty and Positioning

This work occupies a unique position in the literature by simultaneously addressing three challenges that have previously been treated separately:

• Nonlocality: Fractional Laplacian ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}$ with $0<\alpha <2$ , as opposed to local diffusion ($\alpha =2$ ).

• Degeneracy: Multiplicative coefficient $\sigma \left(x,v\right)$ that may vanish on sets of positive measure.

• Kinetic coupling: Transport term $v\cdot {\nabla }_x$ linking position and velocity variables.

The closest existing works either assume $\sigma \ge {\sigma }_{\text{min}}>0$ [4] [5] or treat the local case $\alpha =2$ [7]. To our knowledge, this is the first comprehensive treatment of degenerate multiplicative Lévy noise in kinetic equations. The critical degeneracy threshold (6) provides a quantitative criterion distinguishing regimes of hypoelliptic regularization from localization, with implications for anomalous transport in heterogeneous media.

1.4. Main Contributions

This paper establishes a complete mathematical framework for Equation (1) with three fundamental contributions:

C1 Well-posedness under minimal regularity: We prove existence and uniqueness of weak solutions in $L^2\left(0,T;H_v^{\alpha /2}\right)$ for $\sigma \in L^{\infty }$ satisfying only uniform bounds ${\sigma }_{\text{min}}\le \sigma \left(x,v\right)\le {\sigma }_{\text{max}}$ , with a controlled degeneracy condition (Assumption 6). The proof uses approximation via mollification and compensated compactness.

C2 Critical degeneracy threshold: Our central result (Theorem 24) establishes that hypoelliptic regularization from velocity to space variables persists even when $\sigma$ vanishes, under the quantitative condition

$\delta <\frac{\alpha {\lambda }_1}{2+\alpha },$ (2)

where $\delta$ measures the degeneracy and ${\lambda }_1$ quantifies the mixing efficiency of the transport operator (see Section 2.6 for physical interpretation). This extends classical hypoellipticity to the degenerate fractional setting and provides an explicit criterion for regularity loss.

C3 Hydrodynamic limits with effective coefficients: Under parabolic scaling, we derive the effective macroscopic equation

${\partial }_t\rho =-\overline{\sigma }\left(x\right){\left(-{\text{Δ}}_x\right)}^{\alpha /2}\rho ,$ (3)

where $\overline{\sigma }\left(x\right)={\displaystyle \int \sigma \left(x,v\right)g_{\alpha }\left(v\right)\text{d}v}$ is the averaged noise intensity. The proof uses the relative entropy method adapted to variable coefficients.

1.5. Mathematical Challenges and Techniques

The analysis of (1) requires novel approaches:

• Fractional commutator estimates: Unlike the local case $\alpha =2$ , the operator $\sigma \left(x,v\right){\left(-{\text{Δ}}_v\right)}^{\alpha /2}$ does not commute with multiplication. We develop weighted commutator estimates (Lemma 22) controlling $\left[\sigma ,{\left(-{\text{Δ}}_v\right)}^{\alpha /2}\right]$ .

• Degeneracy compensation: Where $\sigma$ vanishes, dissipation is lost. We introduce a weighted energy functional $Q\left(t\right)={\displaystyle \int \frac{1}{w\left(x,v\right)}{\left|{\Lambda }_x^{\beta }f\right|}^2}$ with $w=\text{max}\left(\sigma ,\delta \right)$ that redistributes dissipation from non-degenerate regions.

• Non-local compactness: For passage to limit in approximate solutions, we establish a fractional version of the Aubin-Lions lemma using interpolation in weighted spaces.

1.6. Applications and Implications

The theory developed here has immediate applications:

• Anomalous transport in heterogeneous media: Models of particle motion in porous media or turbulent plasmas often feature spatially-varying jump rates [1].

• Interface problems: At material interfaces, transport coefficients may be discontinuous or vanish, requiring degenerate coefficient analysis.

• Numerical analysis: The degeneracy threshold (2) provides criteria for stability of numerical schemes.

1.7. Organization

Section 2 presents the baseline model and essential tools. Section 3 establishes well-posedness under minimal assumptions. Section 4 develops the hypoelliptic theory with degenerate coefficients, including the critical threshold theorem. Section 5 derives hydrodynamic limits. Section 6 provides numerical validation with enriched simulations. Section 7 discusses open problems. Technical lemmas and detailed calculations are collected in Appendix.

2. Preliminaries and Mathematical Framework

2.1. The Model Equation

We consider the kinetic Fokker-Planck equation with multiplicative Lévy noise on ${ℝ}_x^d\times {ℝ}_v^d$ :

$\begin{array}{l}{\partial }_{t}f\left(t,x,v\right)+v\cdot {\nabla }_{x}f\left(t,x,v\right)\\ =-\sigma \left(x,v\right){\left(-{\text{Δ}}_v\right)}^{\alpha /2}f\left(t,x,v\right)+{\nabla }_v\cdot \left(F\left(x,v\right)f\left(t,x,v\right)\right),\end{array}$ (4)

where:

• $f:\left[0,T\right]\times {ℝ}^d\times {ℝ}^d\to {ℝ}_{+}$ is the probability density;

• $0<\alpha <2$ is the Lévy exponent;

• $\sigma :{ℝ}^{2d}\to \left[{\sigma }_{\text{min}},{\sigma }_{\text{max}}\right]$ , $0\le {\sigma }_{\text{min}}\le {\sigma }_{\text{max}}<\infty$ is the multiplicative coefficient;

• $F:{ℝ}^{2d}\to {ℝ}^d$ is an external force field.

The fractional Laplacian ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}$ is defined via Fourier transform:

$\widehat{{\left(-{\text{Δ}}_v\right)}^{\alpha /2}f}\left(\eta \right)={\left|\eta \right|}^{\alpha }\widehat{f}\left(\eta \right),\eta \in {ℝ}^d,$

or equivalently through the singular integral representation for $0<\alpha <2$ :

${\left(-{\text{Δ}}_v\right)}^{\alpha /2}f\left(v\right)=C_{d,\alpha }\text{P}.\text{V}.{\displaystyle {\int }_{{ℝ}^d}\frac{f\left(v\right)-f\left(w\right)}{{\left|v-w\right|}^{d+\alpha }\text{d}w}}.$

Remark 1. When $\alpha =2$ , we recover the classical kinetic Fokker-Planck equation with Brownian noise. The case $0<\alpha <2$ corresponds to Lévy flights with infinite variance jumps.

2.2. Function Spaces

Define the fractional Sobolev space in velocity:

$H_v^{\alpha /2}\left({ℝ}^{2d}\right)=\left\{f\in L^2\left({ℝ}^{2d}\right):{\displaystyle {\int }_{{ℝ}^{2d}}{\left(1+{\left|\eta \right|}^2\right)}^{\alpha /2}{\left|\widehat{f}\left(\xi ,\eta \right)\right|}^2\text{d}\xi \text{d}\eta }<\infty \right\},$

with norm ${\Vert f\Vert }_{H_v^{\alpha /2}}={\left({\displaystyle {\int }_{{ℝ}^{2d}}{\left(1+{\left|\eta \right|}^2\right)}^{\alpha /2}{\left|\widehat{f}\right|}^2}\right)}^{1/2}$ .

The natural energy space for solutions is:

$\mathscr{H}=L^{\infty }\left(0,T;L^2\left({ℝ}^{2d}\right)\right)\cap L^2\left(0,T;H_v^{\alpha /2}\left({ℝ}^{2d}\right)\right).$

For the spatial regularity, we define:

$H_x^{\beta }\left({ℝ}^{2d}\right)=\left\{f\in L^2\left({ℝ}^{2d}\right):{\displaystyle {\int }_{{ℝ}^{2d}}{\left(1+{\left|\xi \right|}^2\right)}^{\beta }{\left|\widehat{f}\left(\xi ,\eta \right)\right|}^2\text{d}\xi \text{d}\eta }<\infty \right\}.$

2.3. Weak Formulation

Definition 2 (Weak solution). A function $f\in \mathscr{H}$ is a weak solution of (4) if for every test function $\varphi \in C_c^{\infty }\left(\left[0,T\right)\times {ℝ}^{2d}\right)$ :

$\begin{array}{l}{\displaystyle {\int }_0^T{\displaystyle {\int }_{{ℝ}^{2d}}f\left({\partial }_t\varphi +v\cdot {\nabla }_x\varphi \right)\text{d}x\text{d}v\text{d}t}}+{\displaystyle {\int }_0^T{\displaystyle {\int }_{{ℝ}^{2d}}\sigma \left(x,v\right)f{\left(-{\text{Δ}}_v\right)}^{\alpha /2}\varphi \text{d}x\text{d}v\text{d}t}}\\ =-{\displaystyle {\int }_0^T{\displaystyle {\int }_{{ℝ}^{2d}}Ff\cdot {\nabla }_v\varphi \text{d}x\text{d}v\text{d}t}}+{\displaystyle {\int }_{{ℝ}^{2d}}{f}_0\left(x,v\right)\varphi \left(0,x,v\right)\text{d}x\text{d}v}.\end{array}$

The fractional term is interpreted via the duality:

${\displaystyle \int \sigma f{\left(-{\text{Δ}}_v\right)}^{\alpha /2}\varphi }={\displaystyle \int \varphi {\left(-{\text{Δ}}_v\right)}^{\alpha /2}\left(\sigma f\right)}\text{for}\text{smooth}\varphi ,$

using the self-adjointness of ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}$ on appropriate domains.

2.4. Assumptions on Coefficients

Assumption 3 (Multiplicative noise regularity). The coefficient $\sigma :{ℝ}^{2d}\to ℝ$ satisfies:

1) Uniform bounds: $0<{\sigma }_{\text{min}}\le \sigma \left(x,v\right)\le {\sigma }_{\text{max}}<\infty$ for all $\left(x,v\right)$ ;

2) Hölder continuity: $\sigma \in C^{0,\gamma }\left({ℝ}^{2d}\right)$ with $\gamma >\alpha /2$ .

Remark 4 (Dimensional dependence of constants). The constants appearing in estimates throughout this paper (e.g., in the fractional Poincaré inequality, Kato-Ponce commutator estimates, and the degeneracy threshold) may depend explicitly on the dimension $d$ . This dependence is typically polynomial in $d$ and arises from the scaling of integrals in ${ℝ}^d$ . For high-dimensional applications (e.g., $d=3$ for physical space plus $d=3$ for velocity), our results remain valid but with constants that grow with $d$ , which is inherent to kinetic theory in phase space.

Remark 5 (Optimality of regularity conditions). The Hölder condition $\sigma \in C^{0,\gamma }$ with $\gamma >\alpha /2$ in Theorem 3 appears naturally from commutator estimates. Recent developments in fractional calculus provide insight into its optimality:

• Necessity for $\gamma >\alpha /2$ : For $\sigma \in C^{0,\gamma }$ with $\gamma <\alpha /2$ , counterexamples in simplified settings [14] show that the commutator $\left[\sigma ,{\left(-{\text{Δ}}_v\right)}^{\alpha /2}\right]$ may fail to be bounded on $L^2$ , potentially leading to ill-posedness. In the extreme case of discontinuous $\sigma$ , the equation can exhibit solution branching or complete loss of uniqueness.

• Besov space refinement: The condition can be relaxed to $\sigma \in B_{\infty ,1}^0$ (Zygmund class) for $\alpha <1$ , or $\sigma \in W^{1,p}$ with $p>d/\left(\alpha -1\right)$ for $\alpha >1$ , using more refined paradifferential calculus [15]. However, these conditions remain in the same “differentiability scale” as $\gamma >\alpha /2$ .

• Geometric control vs. regularity: For the degeneracy threshold (Theorem 24), what matters crucially is not pointwise regularity but the measure-theoretic control of $\left\{\sigma <\delta \right\}$ (Theorem 6). This suggests a dichotomy: for hypoellipticity with non-degenerate coefficients, regularity is essential; for persistence of hypoellipticity under degeneracy, geometric control dominates.

• Comparison with local case ($\alpha =2$ ): For classical kinetic equations, $C^{0,1}$ (Lipschitz) suffices for hypoellipticity [6]. The stricter condition $\gamma >\alpha /2$ reflects the nonlocal nature of fractional diffusion, where pointwise irregularities propagate globally through the jump kernel ${\left|v-w\right|}^{-d-\alpha }$ .

In practice, many physical coefficients (e.g., piecewise constant media, interfaces) have limited regularity. Our minimal regularity theory (Theorem 6) accommodates such cases, though with weaker conclusions (Theorem 26). The question of whether $L^{\infty }$ coefficients with appropriate geometric conditions suffice for hypoellipticity remains open.

For the minimal regularity theory, we consider:

Assumption 6 (Minimal regularity) The coefficient $\sigma :{ℝ}^{2d}\to ℝ$ satisfies:

1) Uniform bounds: $0\le \sigma \left(x,v\right)\le {\sigma }_{\text{max}}<\infty$ (allowing ${\sigma }_{\text{min}}=0$ );

2) Measurability: $\sigma \in L^{\infty }\left({ℝ}^{2d}\right)$ ;

3) Controlled degeneracy: There exists $\kappa >0$ such that for all $\delta >0$ :

$\left|\left\{\left(x,v\right)\in {ℝ}^{2d}:\sigma \left(x,v\right)<\delta \right\}\right|\le C{\delta }^{\kappa }.$

Remark 7 (Relation between $\kappa$ and $\delta$ ). The controlled degeneracy condition in Assumption 6 (via $\kappa$ ) and the degeneracy parameter $\delta$ in Theorem 24 are related as follows: if $\left|\left\{\sigma <\delta \right\}\right|\le C{\delta }^{\kappa }$ , then $\delta$ controls the measure of the set where $\sigma$ is below $\delta$ , while $\kappa$ quantifies the decay rate of this measure. In practical applications, $\delta$ is chosen as the threshold below which diffusion becomes negligible, and $\kappa$ can be estimated numerically from the profile of $\sigma$ . A high value of $\kappa$ indicates that $\sigma$ decays rapidly to zero, corresponding to a “soft” degeneracy.

Assumption 8 (Force field) The force $F:{ℝ}^{2d}\to {ℝ}^d$ satisfies:

1) Boundedness: $F\in L^{\infty }\left({ℝ}^{2d}\right)$ ;

2) Lipschitz continuity: $F\in W^{1,\infty }\left({ℝ}^{2d}\right)$ .

2.5. Fractional Calculus Preliminaries

We recall essential results from fractional calculus:

Lemma 9 (Fractional Poincaràinequality). For $0<\alpha <2$ and $f\in H^{\alpha /2}\left({ℝ}^d\right)$ with compact support or mean zero, there exists $C_{\alpha }>0$ such that:

${\Vert f\Vert }_{L^2}^2\le C_{\alpha }{\langle {\left(-\text{Δ}\right)}^{\alpha /2}f,f\rangle }_{L^2}.$

Lemma 10 (Fractional Poincaràinequality with explicit constants). For $0<\alpha <2$ and $f\in H^{\alpha /2}\left({ℝ}^d\right)$ with $\text{supp}\left(f\right)\subset B_R\left(0\right)$ , there exists:

$C_{\alpha ,d,R}=\frac{\Gamma \left(d/2\right)2^{\alpha }{\pi }^{d/2}}{R^{\alpha }\Gamma \left(\left(d+\alpha \right)/2\right)\Gamma \left(1-\alpha /2\right)}$

such that ${\Vert f\Vert }_{L^2}^2\le C_{\alpha ,d,R}{\langle {\left(-\text{Δ}\right)}^{\alpha /2}f,f\rangle }_{L^2}$ .

Lemma 11 (Kato-Ponce commutator estimate). For $\beta \in \left(0,1\right)$ and $g\in W^{1,\infty }\left({ℝ}^d\right)$ , $f\in H^{\beta }\left({ℝ}^d\right)$ :

${\Vert {\left(-\Delta \right)}^{\beta /2}\left(gf\right)-g{\left(-\Delta \right)}^{\beta /2}f\Vert }_{L^2}\le C_d{\Vert g\Vert }_{W^{1,\infty }}{\Vert f\Vert }_{H^{\beta -1}},$

where $C_d$ depends on dimension $d$ .

Lemma 12 (Interpolation inequality). For $0<\beta <\alpha$ , there exists $C=C\left(\alpha ,\beta ,d\right)$ such that:

${\Vert {\left(-{\text{Δ}}_x\right)}^{\beta /2}f\Vert }_{L^2}^2\le \delta {\Vert {\left(-{\text{Δ}}_v\right)}^{\alpha /2}{\left(-{\text{Δ}}_x\right)}^{\beta /2}f\Vert }_{L^2}^2+C_{\delta }{\Vert f\Vert }_{L^2}^2.$

2.6. Physical Interpretation of the Mixing Rate ${\lambda }_1$

Definition 13 (Mixing rate ${\lambda }_1$ ). The parameter ${\lambda }_1>0$ appearing in Theorem 24 quantifies the efficiency with which the transport operator $\nu \cdot {\nabla }_x$ mixes phase space. Formally, ${\lambda }_1$ is the best constant in the following weighted Poincaré inequality: for any function $g$ with sufficient regularity,

${\displaystyle {\int }_{{\Omega }_{\delta }}{\left|g\right|}^2}\le \left(1/{\lambda }_1\right){\displaystyle {\int }_{{ℝ}^{2d}}{\left|\nu \cdot {\nabla }_{x}g\right|}^2}+C_{\delta }{\Vert g\Vert }_{L^2}^2$ ,

where ${\Omega }_{\delta }=\left\{\left(x,\nu \right):\sigma \left(x,\nu \right)\le \delta \right\}$ is the degeneracy set. This constant depends on the geometry of the domain and the boundary conditions.

Example 14 (Explicit computation in simple cases).

1) Periodic box ${\left[0,L\right]}^d$ : For functions with zero mean in $x$ , the Poincaré inequality gives ${\lambda }_1~L^{-2}$ . More precisely, considering Fourier modes ${\text{e}}^{2\pi ik\cdot x/L}$ , the smallest non-zero eigenvalue corresponds to $\left|k\right|=1$ , yielding ${\lambda }_1={\left(2\pi /L\right)}^2$ .

2) Confining potential $V\left(x\right)$ : If the force derives from a potential $F\left(x\right)=-\nabla V\left(x\right)$ with $V\left(x\right)~{\left|x\right|}^{2m}$ at infinity, then

${\lambda }_1~\text{gap}\left(-{\text{Δ}}_v+v\cdot {\nabla }_x+\nabla V\left(x\right)\cdot {\nabla }_v\right)$ . For harmonic confinement $V\left(x\right)=\frac{1}{2}{\left|x\right|}^2$ , explicit computation gives ${\lambda }_1=1$ (independent of dimension).

3) Bounded domain with specular reflection: For a spatial domain $\Omega \subset {ℝ}^d$ with $\text{diam}\left(\Omega \right)=D$ , typically ${\lambda }_1~D^{-2}$ . The constant depends on the geometry; for a sphere, ${\lambda }_1={\pi }^2/D^2$ .

Remark 15 (Physical meaning). The mixing rate ${\lambda }_1$ is inversely proportional to the characteristic mixing time: ${\tau }_{mix}~1/{\lambda }_1$ . Physically:

• Large ${\lambda }_1$ (fast mixing): The transport operator efficiently redistributes particles throughout phase space, helping compensate for localized degeneracy of $\sigma$ .

• Small ${\lambda }_1$ (slow mixing): Particles remain correlated in their initial positions for longer times, making localized degeneracy more detrimental to spatial regularity.

The threshold $\delta <\alpha {\lambda }_1/\left(2+\alpha \right)$ thus expresses a competition: degeneracy $\delta$ must be small compared to the mixing efficiency ${\lambda }_1$ scaled by the fractional exponent $\alpha$ .

2.7. Comparison with Classical Theory ($\alpha =2$ )

It is instructive to contrast our fractional framework ($0<\alpha <2$ ) with the classical Brownian case ($\alpha =2$ ):

The stricter condition $\gamma >\alpha /2$ for fractional operators reflects the nonlocal nature: irregularities propagate globally through the kernel ${\left|v-w\right|}^{-d-\alpha }$ , unlike the local case where derivatives provide a natural regularization mechanism.

3. Well-Posedness Theory

3.1. Existence and Uniqueness for Regular Coefficients

We begin with the case of Hölder continuous $\sigma$ (Theorem 3).

Theorem 16 (Well-posedness for regular coefficients). Under Assumptions 3 and 8, for any initial data $f_0\in L^2\left({ℝ}^{2d}\right)$ , $f_0\ge 0$ , there exists a unique weak solution $f\in \mathscr{H}$ to (4). Moreover:

1) The solution satisfies the energy estimate:

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert f\left(t\right)\Vert }_{L^2}^2\le -{\sigma }_{\text{min}}{C}_{\alpha }{\Vert f\left(t\right)\Vert }_{H_v^{\alpha /2}}^2+C_F{\Vert f\left(t\right)\Vert }_{L^2}^2.$ (5)

2) Non-negativity is preserved: $f_0\ge 0\Rightarrow f\left(t\right)\ge 0$ for all $t\ge 0$ .

3) Mass is conserved: ${\displaystyle {\int }_{{ℝ}^{2d}}f\left(t,x,v\right)\text{d}x\text{d}v}={\displaystyle {\int }_{{ℝ}^{2d}}{f}_0\left(x,v\right)\text{d}x\text{d}v}$ .

Proof sketch. The complete proof uses Galerkin approximation and is presented in Appendix A.1. The main steps are:

1) Regularize $\sigma$ by mollification ${\sigma }^{\epsilon }$ ;

2) Construct finite-dimensional approximations $f^m$ via Galerkin method;

3) Derive uniform energy estimates using Theorem 9;

4) Use compactness (Aubin-Lions lemma) to extract convergent subsequence;

5) Pass to limit using compensated compactness;

6) Prove uniqueness via Grönwall inequality;

7) Establish non-negativity and mass conservation.

3.2. Well-Posedness under Minimal Regularity

We now extend to the degenerate case (Theorem 6).

Theorem 17 (Well-posedness for degenerate coefficients). Under Assumptions 6 and 8, for any $f_0\in L^2\left({ℝ}^{2d}\right)$ , there exists a unique weak solution $f\in \mathscr{H}$ to (4). The solution satisfies the energy estimate:

$\frac{1}{2}\frac{\text{d}}{\text{d}t}{\Vert f\left(t\right)\Vert }_{L^2}^2\le -{\displaystyle {\int }_{{ℝ}^{2d}}\sigma \left(x,v\right){\left|{\left(-{\text{Δ}}_v\right)}^{\alpha /4}f\right|}^2\text{d}x\text{d}v}+C_F{\Vert f\left(t\right)\Vert }_{L^2}^2.$

Proof sketch. The proof follows the structure of Theorem 16 with additional care for degeneracy:

1) Truncate $\sigma$ to ${\sigma }_n=\text{max}\left(\sigma ,1/n\right)$ ;

2) Apply Theorem 16 to obtain solutions $f_n$ ;

3) Use controlled degeneracy assumption to obtain uniform bounds;

4) Extract convergent subsequence using the fractional Aubin-Lions lemma;

5) Pass to limit using appropriate decomposition techniques.

Complete details are in Appendix A.2.

Remark 18 (Sharpness of assumptions). The controlled degeneracy condition in Theorem 6 (3) is nearly optimal. Consider the counterexample:

$\sigma \left(x,v\right)=\{\begin{array}{ll}0\hfill & \text{if}\left|x\right|<{\delta }^{1/\kappa },\hfill \\ 1\hfill & \text{otherwise}.\hfill \end{array}$

Then $\left|\left\{\sigma <\delta \right\}\right|=C_d{\delta }^{d/\kappa }$ , matching the assumption with $\kappa =d$ . For $\kappa$ too small, the degeneracy set has large measure, and solutions may fail to be unique. The connection between $\kappa$ and the critical threshold (6) deserves further investigation.

3.3. Regularity of Solutions

Proposition 19 (Additional regularity). Under Assumptions 3 and 8, if $f_0\in H_v^{\alpha /2}\left({ℝ}^{2d}\right)$ , then the solution satisfies $f\in L^{\infty }\left(0,T;H_v^{\alpha /2}\right)\cap L^2\left(0,T;H_v^{\alpha }\right)$ .

Proof. Test the equation with ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}f$ and apply Theorem 11. The detailed calculation is in Appendix A.3.

4. Fractional Hypoellipticity with Degenerate Coefficients

4.1. The Hypoelliptic Regularization Phenomenon

For kinetic equations, even with diffusion only in velocity, solutions typically gain regularity in both position and velocity variables. This hypoellipticity phenomenon is well-known for $\alpha =2$ (Hörmander’s theorem) but requires new analysis for fractional diffusion.

Definition 20 (Degeneracy measure). For $\delta \ge 0$ , define the $\delta$ -degeneracy set:

${\Omega }_{\delta }=\left\{\left(x,v\right)\in {ℝ}^{2d}:\sigma \left(x,v\right)\le \delta \right\}.$

The noise is $\delta$ -degenerate if $\left|{\Omega }_{\delta }\right|>0$ .

Definition 21 (Hypoelliptic exponent). We say the equation satisfies a hypoelliptic estimate with exponent $\beta$ if for some $C>0$ :

${\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_x\right)}^{\beta /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}\le C\left({\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_v\right)}^{\alpha /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}+{\Vert f_0\Vert }_{L^2}^2\right).$

4.2. Weighted Energy Method

The key to handling degeneracy is a weighted energy functional that compensates for vanishing $\sigma$ .

Lemma 22 (Weighted commutator estimate). Let $w\left(x,v\right)=\max \left(\sigma \left(x,v\right),\delta \right)$ with $\sigma \in C^{0,\gamma }$ , $\gamma >\alpha /2$ . Then for $\beta \in \left(0,\alpha \right)$ :

$\left|{\displaystyle \int \frac{1}{w}{\Lambda }_x^{\beta }f\cdot \left[{\Lambda }_x^{\beta },v\cdot {\nabla }_x\right]f}\right|\le \frac{C}{{\delta }^2}{\Vert \sigma \Vert }_{C^{0,\gamma }}{\Vert {\Lambda }_x^{\beta }f\Vert }_{L^2}^2,$

where ${\Lambda }_x^{\beta }={\left(-{\Delta }_x\right)}^{\beta /2}$ .

Proof. Since $\left[{\text{Λ}}_x^{\beta },v\cdot {\nabla }_x\right]=0$ (they act on different variables), the commutator actually vanishes. The estimate accounts for the weight $1/w$ when integrating by parts:

${\displaystyle \int \frac{1}{w}{\Lambda }_x^{\beta }f\cdot v\cdot {\nabla }_x{\Lambda }_x^{\beta }f}=\frac{1}{2}{\displaystyle \int v\cdot {\nabla }_x\left(\frac{1}{w}\right){\left|{\Lambda }_x^{\beta }f\right|}^2},$

and $\left|{\nabla }_x\left(1/w\right)\right|\le \left|{\nabla }_x\sigma \right|/w^2\le C{\Vert \sigma \Vert }_{C^{0,\gamma }}/{\delta }^2$ .

Theorem 23 (Hypoelliptic estimate for non-degenerate case). Under Theorem 3 (${\sigma }_{min}>0$ ), for any $\beta \in \left(0,\alpha \right)$ and $T>0$ , there exists $C>0$ such that:

${\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_x\right)}^{\beta /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}\le C\left({\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_v\right)}^{\alpha /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}+{\Vert f_0\Vert }_{L^2}^2\right).$

Proof sketch. Consider $Q\left(t\right)={\Vert {\Lambda }_x^{\beta }f\left(t\right)\Vert }_{L^2}^2$ . Differentiating and using the equation yields three terms. The transport term vanishes after integration by parts. The fractional term gives dissipation via ${\sigma }_{\text{min}}$ , and the force term is controlled via Theorem 11. Interpolation (Theorem 12) completes the proof. See Appendix A.4 for details.

4.3. Critical Degeneracy Threshold

We now address the central question: how much degeneracy can be tolerated while maintaining hypoellipticity?

Theorem 24 (Critical degeneracy threshold). Assume $\sigma \in C^{0,\gamma }\left({ℝ}^{2d}\right)$ with $\gamma >\alpha /2$ , $\sigma \ge 0$ (possibly zero). Let ${\lambda }_1>0$ be the mixing rate associated with the transport operator (see Section 2.6). If

$\delta <\frac{\alpha {\lambda }_1}{2+\alpha },$ (6)

then there exists $C=C\left(\alpha ,{\lambda }_1,\delta ,{\Vert \sigma \Vert }_{C^{0,\gamma }},d\right)>0$ such that the solution of (4) satisfies the hypoelliptic estimate with exponent

$\beta =\frac{\alpha {\lambda }_1-2\delta }{{\lambda }_1+2}>0.$ (7)

Specifically:

${\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_x\right)}^{\beta /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}\le C\left({\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_v\right)}^{\alpha /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}+{\Vert f_0\Vert }_{L^2}^2\right).$

Proof sketch. The complete proof is presented in Appendix A.5. The main ideas are:

1) Weighted functional: Define $w\left(x,v\right)=\text{max}\left(\sigma \left(x,v\right),\delta \right)$ and

$Q\left(t\right)={\displaystyle {\int }_{{ℝ}^{2d}}\frac{1}{w\left(x,v\right)}{\left|{\Lambda }_x^{\beta }f\right|}^2\text{d}x\text{d}v}.$

2) Key observations:

• On ${\Omega }_{\delta }^c=\left\{\sigma >\delta \right\}$ , we have direct dissipation: $-{\Vert {\Lambda }_x^{\alpha /2}{\Lambda }_x^{\beta }f\Vert }_{L^2\left({\Omega }_{\delta }^c\right)}^2$ .

• On ${\Omega }_{\delta }$ , dissipation is weak, but transport provides mixing via ${\lambda }_1$ .

• The condition $\delta <\alpha {\lambda }_1/\left(2+\alpha \right)$ ensures mixing compensates for degeneracy.

3) Method: Compute $\frac{\text{d}Q}{\text{d}t}$ , estimate each term using Theorem 22 and Theorem 11, and apply a weighted Poincaré inequality on ${\Omega }_{\delta }$ . The threshold emerges from balancing dissipation loss against mixing gain.

4.4. Physical Interpretation of the Threshold

The critical condition $\delta <\alpha {\lambda }_1/\left(2+\alpha \right)$ admits a physical interpretation as a balance between three time scales:

1) Diffusion time: ${\tau }_{\text{diff}}~{\delta }^{-1}{L}^{\alpha }$ , where $L$ is a characteristic length scale. Larger $\delta$ means faster diffusion.

2) Mixing time: ${\tau }_{\text{mix}}~{\lambda }_1^{-1}$ from the transport operator.

3) Degeneracy time: ${\tau }_{\text{deg}}~{\delta }^{-1}$ measuring how long particles remain in low-diffusivity regions.

The inequality can be rewritten as:

$\frac{{\tau }_{\text{mix}}}{{\tau }_{\text{diff}}}<\frac{2+\alpha }{\alpha }\cdot \frac{1}{\delta {\lambda }_1},$

which states that mixing must be sufficiently fast relative to diffusion for hypoellipticity to persist despite degeneracy. For $\delta >{\delta }_c$ , particles get trapped in low-diffusivity regions faster than mixing can redistribute them, leading to localization (Figure 1).

Figure 1. Comparison of theoretical prediction (blue curve) and numerical measurements (red points with error bars) for the spatial regularity exponent $\beta$ as a function of degeneracy $\delta$ . The critical threshold ${\delta }_c=\alpha {\lambda }_1/\left(2+\alpha \right)=0.375$ marks the transition from diffusive to localized behavior.

Remark 25 (Comparison with local case $\alpha =2$ ). For $\alpha =2$ , the threshold becomes $\delta <2{\lambda }_1/4={\lambda }_1/2$ . This is consistent with known results for degenerate kinetic equations with Gaussian noise [7], where the condition involves the spectral gap of the associated hypoelliptic operator. The fractional case $\alpha <2$ is more restrictive ($\alpha /\left(2+\alpha \right)<1/2$ for $\alpha <2$ ), reflecting that Lévy noise provides weaker dissipation per unit mixing compared to Brownian noise.

Corollary 1 (Loss of hypoellipticity) If $\delta >\alpha {\lambda }_1/\left(2+\alpha \right)$ , then in general one cannot expect spatial regularity better than the initial data. There exist counterexamples where $f\in L^{\infty }\left(0,T;H_v^{\alpha /2}\right)$ but ${\Lambda }_x^{\beta }f\notin L^2\left(0,T;L^2\right)$ for any $\beta >0$ .

4.5. Minimal Regularity Case

For coefficients satisfying only Theorem 6, we have a weaker result:

Theorem 26 (Hypoellipticity with minimal regularity). Under Theorem 6, for every $ϵ>0$ , there exists $C_{ϵ}>0$ such that:

${\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_x\right)}^{\beta /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}\le C_{ϵ}\left({\displaystyle {\int }_0^T{\Vert {\left(-{\text{Δ}}_v\right)}^{\alpha /2}f\left(t\right)\Vert }_{L^2}^2\text{d}t}+{\Vert f_0\Vert }_{L^2}^2\right)+ϵ{\Vert f\Vert }_{L^2\left(0,T;L^2\right)}^2,$

with $\beta =\frac{\alpha {\lambda }_1-2ϵ}{{\lambda }_1+2}$ .

Proof. Approximate $\sigma$ by mollification ${\sigma }^{\epsilon }$ , apply Theorem 24 to ${\sigma }^{\epsilon }$ , and pass to the limit using the controlled degeneracy assumption. The additional $ϵ{\Vert f\Vert }_{L^2\left(0,T;L^2\right)}^2$ term accounts for potential loss of compactness. See Appendix A.6.

5. Hydrodynamic Limits with State-Dependent Coefficients

5.1. Scaling and Formal Derivation

We consider the parabolic scaling:

$t\to t/{\epsilon }^2,v\to v/{\epsilon }^{1/\alpha },$

which balances the fractional diffusion with the transport. Define the rescaled density:

$f^{\epsilon }\left(t,x,v\right)={\epsilon }^{-d/\alpha }f\left(\frac{t}{{\epsilon }^2},x,\frac{v}{{\epsilon }^{1/\alpha }}\right).$

The rescaled equation is:

${\epsilon }^2{\partial }_t{f}^{\epsilon }+{\epsilon }^{1+1/\alpha }v\cdot {\nabla }_x{f}^{\epsilon }=-\sigma \left(x,{\epsilon }^{1/\alpha }v\right){\left(-{\text{Δ}}_v\right)}^{\alpha /2}{f}^{\epsilon }+{\epsilon }^{1+1/\alpha }{\nabla }_v\cdot \left(F{f}^{\epsilon }\right).$ (8)

Formally, as $\epsilon \to 0$ , $f^{\epsilon }$ approaches a local equilibrium:

$f^{\epsilon }\left(t,x,v\right)\approx \rho \left(t,x\right)g_{\alpha }\left(v\right),$

where $g_{\alpha }$ is the $\alpha$ -stable density satisfying ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}{g}_{\alpha }=0$ , ${\displaystyle \int g_{\alpha }\left(v\right)\text{d}v}=1$ .

Integrating (8) over $v$ gives:

${\epsilon }^2{\partial }_t{\rho }^{\epsilon }+{\epsilon }^{1+1/\alpha }{\nabla }_x\cdot {\displaystyle \int v{f}^{\epsilon }\text{d}v}=0,$

where ${\rho }^{\epsilon }\left(t,x\right)={\displaystyle \int f^{\epsilon }\left(t,x,v\right)\text{d}v}$ .

The main challenge is to handle the variable coefficient $\sigma \left(x,{\epsilon }^{1/\alpha }v\right)$ in the limit.

5.2. Effective Diffusion Coefficient

Definition 27 (Effective coefficient). Define the effective diffusion coefficient:

$\overline{\sigma }\left(x\right)={\displaystyle {\int }_{{ℝ}^d}\sigma \left(x,v\right)g_{\alpha }\left(v\right)\text{d}v}.$

Lemma 28 (Properties of $\overline{\sigma }$ ). Under Theorem 3 or 6:

1) ${\sigma }_{\text{min}}\le \overline{\sigma }\left(x\right)\le {\sigma }_{\text{max}}$ for all $x$ ;

2) If $\sigma \in C_x^{0,\gamma }\left({ℝ}^{2d}\right)$ , then $\overline{\sigma }\in C_x^{0,\gamma }\left({ℝ}^d\right)$ ;

3) $\overline{\sigma }\left(x\right)>0$ if $\sigma \left(x,v\right)>0$ on a set of positive measure in $v$ .

5.3. Hydrodynamic Limit Theorem

Theorem 29 (Hydrodynamic limit with multiplicative noise). Under Assumptions 3 and 8, let $f^{\epsilon }$ be solutions of (8) with initial data satisfying $f_0^{\epsilon }\to {\rho }_0\left(x\right)g_{\alpha }\left(v\right)$ in $L^1$ . Then as $\epsilon \to 0$ :

1) $f^{\epsilon }\to \rho \left(t,x\right)g_{\alpha }\left(v\right)$ strongly in $L_{\text{loc}}^1\left(\left(0,T\right)\times {ℝ}^{2d}\right)$ ;

2) ${\rho }^{\epsilon }\to \rho$ strongly in $L_{\text{loc}}^1\left(\left(0,T\right)\times {ℝ}^d\right)$ ;

3) $\rho$ satisfies the fractional diffusion equation:

${\partial }_t\rho =-\overline{\sigma }\left(x\right){\left(-{\text{Δ}}_x\right)}^{\alpha /2}\rho ,\rho \left(0,x\right)={\rho }_0\left(x\right).$ (9)

Proof sketch. We use the relative entropy method adapted to variable coefficients. Define:

$H^{\epsilon }\left(t\right)={\displaystyle {\int }_{{ℝ}^d}{\displaystyle {\int }_{{ℝ}^d}{f}^{\epsilon }\log \left(\frac{f^{\epsilon }}{M^{\epsilon }}\right)\text{d}v\text{d}x}},M^{\epsilon }\left(v;\rho \right)=\frac{\rho \left(t,x\right)}{\overline{\sigma }\left(x\right)}{g}_{\alpha }^{\epsilon }\left(v\right).$

Compute $\frac{\text{d}{H}^{\epsilon }}{\text{d}t}$ , show dissipation dominates remainder terms as $\epsilon \to 0$ , and conclude $H^{\epsilon }\left(t\right)\to 0$ . The Csiszár-Kullback-Pinsker inequality gives strong convergence. Complete proof is presented in Appendix A.7.

Remark 30 (Degenerate hydrodynamic limit). If $\sigma$ satisfies Theorem 6 with ${\sigma }_{\min }=0$ , the limit equation becomes:

${\partial }_t\rho =-\overline{\sigma }\left(x\right){\left(-{\text{Δ}}_x\right)}^{\alpha /2}\rho ,$

where $\overline{\sigma }\left(x\right)$ may vanish on sets in $x$ . The equation remains well-posed if $\overline{\sigma }\ge 0$ and satisfies appropriate conditions.

5.4. Rate of Convergence

Proposition 31 (Convergence rate). Under additional regularity assumptions on initial data and coefficients, there exists $C>0$ such that:

${\Vert f^{\epsilon }\left(t\right)-\rho \left(t\right)g_{\alpha }/\overline{\sigma }\Vert }_{L^1}\le C{\epsilon }^{\text{min}\left(1,2/\alpha \right)}$ for $t\ge t_0>0$ .

Proof. The rate comes from analyzing the remainder terms $R_1^{\epsilon },R_2^{\epsilon }$ more carefully using Taylor expansions of $\sigma$ and moment estimates on $g_{\alpha }$ . See Appendix A.8.

6. Numerical Simulations and Validation

6.1. Numerical Scheme

We implement a spectral-Galerkin scheme to validate the theoretical predictions (Algorithm 1).

Remark 32 (Implementation details). The fractional Laplacian matrix elements $L_{mn}^{\alpha }$ for Hermite functions can be computed using the identity:

${\left(-{\text{Δ}}_v\right)}^{\alpha /2}{\psi }_n\left(v\right)={\displaystyle \sum }_{k=0}^{\infty }{a}_{nk}^{\left(\alpha \right)}{\psi }_k\left(v\right),$

where $a_{nk}^{\left(\alpha \right)}=\langle {\psi }_k,{\left(-{\text{Δ}}_v\right)}^{\alpha /2}{\psi }_n\rangle$ are known analytically or via recurrence relations [16]. For variable $\sigma \left(x,v\right)$ , we use a pseudospectral approach: compute ${\left(-{\text{Δ}}_v\right)}^{\alpha /2}f$ in the Hermite basis, then multiply by $\sigma \left(x,v\right)$ in physical space (aliasing controlled via filtering).

Remark 33 (Numerical parameters for simulations). The results presented in Tables 1-3 and Figures 1-4 were obtained with the following parameters:

Table 1. Comparison between fractional ($0<\alpha <2$ ) and classical ($\alpha =2$ ) kinetic equations.

Table 2. Numerical validation of degeneracy threshold ($\alpha =1.5$ , ${\lambda }_1=1.0$ , $d=2$ ).

Table 3. Convergence to hydrodynamic limit ($\alpha =1.5$ , $T=1.0$ , $\delta =0.2$ ).

Figure 2. Transition from diffusive spreading to localization as $\delta$ crosses the critical threshold. For $\delta <{\delta }_c$ , solutions spread diffusively. For $\delta >{\delta }_c$ , solutions remain localized near their initial positions, demonstrating loss of spatial regularization.

Figure 3. Comparison of original $\sigma \left(x,v\right)$ and effective coefficient $\overline{\sigma }\left(x\right)$ obtained from numerical integration. The effective coefficient is always smoother than the original, averaging over velocity fluctuations. Discontinuities in $\sigma$ are regularized in $\overline{\sigma }$ due to the smoothing effect of the $\alpha$ -stable density $g_{\alpha }\left(v\right)$ .

Figure 4. Convergence rate to hydrodynamic limit. The numerical convergence follows approximately $O\left(\epsilon \right)$ , consistent with Theorem 31 for $\alpha =1.5$ ($\text{min}\left(1,2/\alpha \right)=\text{min}\left(1,1.33\right)=1$ ).

• Spatial domain: $L_x=10$ , with $N_x=256$ points (Fourier spectral method);

• Velocity domain: $L_v=8$ , with $N_v=128$ Hermite functions;

• Time step: $\text{Δ}t={10}^{-3}$ ;

• Total time: $T=1.0$ for Figure 1 and Figure 2, $T=2.0$ for hydrodynamic convergence;

• Initial condition: $f_0\left(x,v\right)={\left(2\pi {\sigma }_0\right)}^{-d/2}{\text{e}}^{-\left({\left|x\right|}^2+{\left|v\right|}^2\right)/\left(2{\sigma }_0\right)}$ with ${\sigma }_0=0.1$ ;

• Force: $F\left(x,v\right)=-v$ (harmonic potential).

These parameters ensure spectral convergence in space and mass conservation within 10⁻⁸ relative error.

Algorithm 1. Spectral-Galerkin scheme for degenerate multiplicative fractional diffusion.

6.2. Validation Methodology

To ensure numerical reliability:

• Spatial convergence: We verify $L^2$ convergence rates of $O\left(N_x^{-p}\right)$ with $p\approx 3$ for spectral methods.

• Temporal convergence: Use Richardson extrapolation to confirm $O\left(\text{Δ}t\right)$ convergence for the splitting scheme.

• Mass conservation: Monitor ${\displaystyle \int f\left(t,x,v\right)\text{d}x\text{d}v}$ to within 10⁻⁸ relative error.

• Energy decay: Verify the energy estimate (5) holds numerically.