A unified fractional-stochastic energy balance model: Theoretical analysis, numerical implementation, and implications for sustainable energy systems

Modeling climate-energy interactions requires frameworks that capture both long-range memory and extreme event risks. Classical energy balance models (EBMs) are limited by their memoryless deterministic structure, while purely stochastic extensions typically incorporate only Gaussian noise and fail to account for heavy-tailed extremes. This paper introduces a unified fractional-stochastic energy balance equation (FS-EBE) that combines Caputo fractional integrals for memory effects with Lévy processes for stochastic forcing, including jumps. We establish the model's well-posedness by proving existence, uniqueness, and moment boundedness for the resulting stochastic Volterra equation. A dedicated fractional-stochastic Adams-Bashforth-Moulton (ABM) predictor-corrector scheme is developed for a numerical solution. Through two case studies—periodic solar forcing with Gaussian noise and correlated forcing with Lévy jumps—we demonstrate that the FS-EBE produces phase lags, long-tailed relaxation following extremes, and heavier-tailed anomaly distributions absent in classical and standard stochastic EBMs. Two novel indices are introduced: the fractional memory index ( $M=1-\alpha$) quantifying system memory and the stochastic fractional robustness index (SFRI) measuring finite-time resilience. These provide quantitative tools for assessing energy infrastructure vulnerability to climate extremes, supporting Sustainable Development Goal 7 affortable and clean energy.