Fractional Risk Analysis of Stochastic Systems with Jumps and Memory

Accurate risk assessment is essential for safety-critical autonomous and control systems under uncertainty. In many real-world settings, stochastic dynamics exhibit asymmetric jumps and long-range memory, making long-term risk probabilities difficult to estimate across varying system dynamics, initial conditions, and time horizons. Existing sampling-based methods are computationally expensive due to repeated long-horizon simulations to capture rare events, while existing partial differential equation (PDE)-based formulations are largely limited to Gaussian or symmetric jump dynamics and typically treat memory effects in isolation. In this paper, we address these challenges by deriving a space- and time-fractional PDE that characterizes long-term safety and recovery probabilities for stochastic systems with both asymmetric Levy jumps and memory. This unified formulation captures nonlocal spatial effects and temporal memory within a single framework and enables the joint evaluation of risk across initial states and horizons. We show that the proposed PDE accurately characterizes long-term risk and reveals behaviors that differ fundamentally from systems without jumps or memory and from standard non-fractional PDEs. Building on this characterization, we further demonstrate how physics-informed learning can efficiently solve the fractional PDEs, enabling accurate risk prediction across diverse configurations and strong generalization to out-of-distribution dynamics.