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Beyond Gaussian Assumptions: A General Fractional HJB Control Framework for Lévy-Driven Heavy-Tailed Channels in 6G

Mengqi Li, Lixin Li, Wensheng Lin, Zhu Han, Tamer Başar

TL;DR

The paper tackles non-Gaussian, heavy-tailed fading in 6G HRLLC by introducing a symmetric $α$-stable Lévy-driven channel model and a generalized fractional HJB control framework using the $\nabla_σ^α$ (Riesz) operator. It derives a continuous-time, slow–fast state-space model that captures both long-term and short-term fading and formulates a fractional HJB equation, ∂_t V + H(∇_x V) + ∇^α_σ V = 0, with terminal data and proves existence and uniqueness of viscosity solutions via Perron’s method and a comparison principle. The framework is validated through numerical simulations in a multi-cell, multi-user downlink, showing that the Lévy-driven controller learns coordinated, robust power-control policies that mitigate impulsive interference and maintain SINR near a target threshold, outperforming Gaussian benchmarks in terms of resilience to jumps. The results underscore the practical potential of fractional-order, non-Gaussian control for 6G communications in dynamic, interference-limited environments.

Abstract

Emerging 6G wireless systems suffer severe performance degradation in challenging environments like high-speed trains traversing dense urban corridors and Unmanned Aerial Vehicles (UAVs) links over mountainous terrain. These scenarios exhibit non-Gaussian, non-stationary channels with heavy-tailed fading and abrupt signal fluctuations. To address these challenges, this paper proposes a novel wireless channel model based on symmetric $α$-stable Lévy processes, thereby enabling continuous-time state-space characterization of both long-term and short-term fading. Building on this model, a generalized optimal control framework is developed via a fractional Hamilton-Jacobi-Bellman (HJB) equation that incorporates the Riesz fractional operator to capture non-local spatial effects and memory-dependent dynamics. The existence and uniqueness of viscosity solutions to the fractional HJB equation are rigorously established, thus ensuring the theoretical validity of the proposed control formulation. Numerical simulations conducted in a multi-cell, multi-user downlink setting demonstrate the effectiveness of the fractional HJB-based strategy in optimizing transmission power under heavy-tailed co-channel and multi-user interference.

Beyond Gaussian Assumptions: A General Fractional HJB Control Framework for Lévy-Driven Heavy-Tailed Channels in 6G

TL;DR

The paper tackles non-Gaussian, heavy-tailed fading in 6G HRLLC by introducing a symmetric -stable Lévy-driven channel model and a generalized fractional HJB control framework using the (Riesz) operator. It derives a continuous-time, slow–fast state-space model that captures both long-term and short-term fading and formulates a fractional HJB equation, ∂_t V + H(∇_x V) + ∇^α_σ V = 0, with terminal data and proves existence and uniqueness of viscosity solutions via Perron’s method and a comparison principle. The framework is validated through numerical simulations in a multi-cell, multi-user downlink, showing that the Lévy-driven controller learns coordinated, robust power-control policies that mitigate impulsive interference and maintain SINR near a target threshold, outperforming Gaussian benchmarks in terms of resilience to jumps. The results underscore the practical potential of fractional-order, non-Gaussian control for 6G communications in dynamic, interference-limited environments.

Abstract

Emerging 6G wireless systems suffer severe performance degradation in challenging environments like high-speed trains traversing dense urban corridors and Unmanned Aerial Vehicles (UAVs) links over mountainous terrain. These scenarios exhibit non-Gaussian, non-stationary channels with heavy-tailed fading and abrupt signal fluctuations. To address these challenges, this paper proposes a novel wireless channel model based on symmetric -stable Lévy processes, thereby enabling continuous-time state-space characterization of both long-term and short-term fading. Building on this model, a generalized optimal control framework is developed via a fractional Hamilton-Jacobi-Bellman (HJB) equation that incorporates the Riesz fractional operator to capture non-local spatial effects and memory-dependent dynamics. The existence and uniqueness of viscosity solutions to the fractional HJB equation are rigorously established, thus ensuring the theoretical validity of the proposed control formulation. Numerical simulations conducted in a multi-cell, multi-user downlink setting demonstrate the effectiveness of the fractional HJB-based strategy in optimizing transmission power under heavy-tailed co-channel and multi-user interference.

Paper Structure

This paper contains 27 sections, 7 theorems, 77 equations, 7 figures, 2 tables.

Table of Contents

  1. Introduction
  2. System Model
  3. Fading Structure and Power Evolution
  4. Classical Fading Mechanisms
  5. Long-Term Fading
  6. Short-Term Fading
  7. $\alpha$-Stable Distributions
  8. Wireless Channel Modeling with Lévy-driven Fractional Dynamics
  9. Non-Gaussian Long-term Fading
  10. Non-Gaussian Short-term Fading
  11. In-Phase and Quadrature Components
  12. Signal Envelope and Power
  13. Dynamics of the Instantaneous Power
  14. The Composite Channel Model
  15. The Non-Gaussian long-term Fading Process
  16. ...and 12 more sections

Key Result

Lemma 1

Let $X(t)$ be a real-valued càdlàg semimartingale with jumps, and let $f \in C^2(\mathbb{R})$ be a twice continuously differentiable function. Then, the process $Y(t) = f(X(t))$ satisfies where $X_{t-}$ denotes the left-limit of $X$ at time $t$ (well-defined since $X$ is càdlàg). $\Delta X_s = X_s - X_{s-}$ is the jump size of $X$ at time $s$. $[X]^c_t$ is the quadratic variation of the continuou

Figures (7)

  • Figure 1: Illustration of signal components, transmitted at different moments, arriving simultaneously at the UE at moment $\tilde{t}^+$.
  • Figure 2: System Flow Chart.
  • Figure 3: Evolution of the BS Transmit Power and Control Actions after DP Iterations with $\alpha=1.8$.
  • Figure 4: Evolution of the Value Function and SINR Trajectories after DP Iterations with $\alpha=1.8$.
  • Figure 5: Comparison between Received Power $p_{i\ell}(t)$ and Stochastic Noise Term under Different Noise Intensities
  • ...and 2 more figures

Theorems & Definitions (10)

  • Lemma 1: Generalized Itô Formula for Jump Semimartingales SDE
  • Lemma 3: Closure of S$\alpha$S Distributions under Scaling and Linear Combinations Samorodnitsky
  • Definition 1: Viscosity Subsolution
  • Definition 2: Viscosity Supersolution
  • Definition 3: Viscosity Solution
  • Lemma 4
  • Theorem 1: Existence of Viscosity Solution
  • Theorem 2: Comparison Principle
  • Corollary 1: Uniqueness of Viscosity Solution
  • Theorem 3: Existence and Uniqueness of Viscosity Solution