arxiv: 2603.26009 · v2 · submitted 2026-03-27 · 📡 eess.SY · cs.SY· math.OC

Recognition: no theorem link

Fractional Risk Analysis of Stochastic Systems with Jumps and Memory

Yimeng Sun , Zhuoyuan Wang , Xiaole Zhang , Heng Ping , Jintang Xue , Paul Bogdan , Yorie Nakahira

Authors on Pith no claims yet

Pith reviewed 2026-05-14 23:32 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords fractional PDELevy jumpsstochastic riskmemory effectssafety probabilitiesphysics-informed learningrecovery probabilities

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The pith

A space- and time-fractional PDE characterizes long-term safety and recovery probabilities for stochastic systems with asymmetric Levy jumps and memory.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper derives a unified fractional PDE that incorporates both the nonlocal spatial effects of asymmetric Levy jumps and the temporal memory in stochastic system dynamics. This single equation yields safety and recovery probabilities jointly over ranges of initial states and time horizons, avoiding the need for repeated long-horizon Monte Carlo runs. The formulation produces risk behaviors distinct from those of Gaussian or memoryless systems and from standard non-fractional PDEs. Physics-informed learning then solves the PDE efficiently, supporting accurate predictions even for out-of-distribution dynamics.

Core claim

A space- and time-fractional partial differential equation is derived that governs the long-term safety and recovery probabilities of stochastic processes possessing both asymmetric Levy jumps and memory. The equation simultaneously encodes nonlocal spatial jump effects through a fractional space operator and temporal memory through a fractional time operator, permitting direct evaluation of risk quantities across arbitrary initial conditions and horizons within one framework. The resulting probabilities differ systematically from those obtained under Gaussian noise, symmetric jumps, or Markovian dynamics, and the PDE admits accurate numerical solution via physics-informed neural networks.

What carries the argument

The space- and time-fractional PDE whose fractional derivatives in space capture asymmetric Levy jump statistics and whose fractional time derivative encodes memory, thereby unifying the two effects into joint risk probabilities.

If this is right

Risk assessment for systems with jumps and memory can be performed once for an entire family of initial states and horizons rather than through separate simulations.
Physics-informed solvers trained on the fractional PDE generalize to dynamics outside the training distribution.
Risk metrics become available without isolating jump effects from memory effects.
The same PDE framework supplies both safety probabilities and recovery probabilities under one operator.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The fractional formulation may extend naturally to controlled systems where the derived PDE informs feedback design for risk reduction.
Similar joint risk evaluation could apply to other non-Markovian processes whose generators admit fractional representations.
Numerical efficiency gains from the PDE approach may enable online risk monitoring in autonomous systems subject to rare but heavy-tailed disturbances.

Load-bearing premise

The fractional PDE obtained by extending the generator of the jump-memory process exactly reproduces the true long-term safety and recovery probabilities of the underlying stochastic dynamics.

What would settle it

Monte Carlo simulation of the original stochastic differential equation with Levy jumps and memory over long horizons, followed by direct numerical comparison of the empirical safety and recovery frequencies against the values predicted by the fractional PDE at matching initial states and times.

Figures

Figures reproduced from arXiv: 2603.26009 by Heng Ping, Jintang Xue, Paul Bogdan, Xiaole Zhang, Yimeng Sun, Yorie Nakahira, Zhuoyuan Wang.

**Figure 1.** Figure 1: Recovery probability with and without Levy jumps. ´ recovery probability of systems with both jumps and memory, the proof entails that the effect of jumps and memory can be separately analyzed, resulting in space- and timefractional terms in the PDE characterization, respectively. This is verified in the experiments in Section V-A. V. EXPERIMENTS This section presents several experimental results demons… view at source ↗

**Figure 3.** Figure 3: Safety probability for 2D system with OOD dynamics. time instances. The results show that the model can produce accurate predictions even for OOD dynamics (relative L 2 error 2.37×10−2 ), which are fundamentally challenging for existing sampling-based methods (Table I). VI. CONCLUSIONS In this paper, we studied long-term safety and recovery probabilities for stochastic dynamical systems with jumps and memo… view at source ↗

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives a space-time fractional PDE for long-term risk in Levy-jump systems with memory and pairs it with a physics-informed solver, but the generator-to-PDE step is not shown or verified in the supplied text.

read the letter

The main takeaway is a claimed derivation of a combined space- and time-fractional PDE that gives safety and recovery probabilities for stochastic dynamics with asymmetric Levy jumps plus memory, solved via physics-informed learning to avoid heavy Monte Carlo runs across states and horizons. This unification is presented as filling a gap left by prior PDE methods that handled only Gaussian or symmetric jumps and kept memory separate. If the steps hold, it could be useful for control systems where both features matter for rare-event bounds. The motivation section does a clear job stating why sampling is expensive for long horizons and why existing fractional approaches fall short on the joint case. The learning angle is a reasonable practical step for evaluating the PDE over ranges of initial conditions. The soft spot is exactly where the stress-test note lands: the abstract asserts that the PDE accurately characterizes the probabilities, yet the text gives no explicit calculation showing that the fractional Laplacian reproduces the integral jump operator for a general Levy measure or that the time-fractional term matches the memory kernel. Without that equivalence or error bounds, the risk numbers rest on an unverified link. No numerical metrics or out-of-distribution tests appear either, so the generalization claim stays untested here. This is for people working on safety certification in autonomous systems with non-Markovian jump noise. A reader who needs a candidate framework for joint jump-memory risk would find the idea worth checking, provided the derivation is supplied and checked. It is worth sending to peer review so the central step can be examined in full.

Referee Report

2 major / 2 minor

Summary. The paper derives a space- and time-fractional PDE that is asserted to exactly characterize long-term safety and recovery probabilities for stochastic systems driven by asymmetric Lévy jumps together with memory kernels. It further develops a physics-informed learning procedure to solve the resulting fractional PDEs numerically, claiming that this yields accurate risk predictions across initial states and horizons with improved generalization relative to sampling-based or non-fractional PDE methods.

Significance. If the generator-to-PDE equivalence is rigorously established, the work would supply a unified nonlocal framework for risk quantification in systems that simultaneously exhibit jump discontinuities and long-range temporal dependence—features common in autonomous control applications. The physics-informed solver could then enable scalable evaluation of safety metrics that are otherwise intractable by direct simulation, particularly for rare-event probabilities over long horizons.

major comments (2)

[PDE derivation (Section 3)] The central claim rests on the assertion that the derived space-time fractional PDE is equivalent to the infinitesimal generator of the underlying Lévy-driven process with memory. No explicit calculation is supplied showing that the fractional Laplacian term reproduces the integral jump operator for a general asymmetric Lévy measure, nor that the chosen time-fractional derivative (Caputo or Riemann-Liouville) exactly recovers the memory kernel. This equivalence is load-bearing for all subsequent risk predictions.
[Numerical results and validation (Section 5)] The abstract and summary state that the PDE “accurately characterizes” long-term risk, yet the supplied text contains no error metrics, convergence rates, or direct Monte-Carlo comparisons that would quantify the discrepancy between solutions of the fractional PDE and the true safety/recovery probabilities of the original jump process. Without such verification, the generalization claims for the physics-informed learner cannot be assessed.

minor comments (2)

[Preliminaries] Notation for the fractional orders α (space) and β (time) should be introduced with explicit admissible ranges and any constraints arising from the Lévy measure or memory kernel.
[Figures 2–4] Figure captions and axis labels in the risk-surface plots should explicitly state the values of the fractional orders and the specific Lévy measure used, to allow readers to reproduce the qualitative differences claimed versus the non-fractional case.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and the opportunity to clarify the manuscript. We address both major points by expanding the derivation with explicit calculations and by adding quantitative validation metrics and Monte-Carlo comparisons. These revisions will make the generator-to-PDE equivalence fully transparent and will substantiate the accuracy and generalization claims.

read point-by-point responses

Referee: [PDE derivation (Section 3)] The central claim rests on the assertion that the derived space-time fractional PDE is equivalent to the infinitesimal generator of the underlying Lévy-driven process with memory. No explicit calculation is supplied showing that the fractional Laplacian term reproduces the integral jump operator for a general asymmetric Lévy measure, nor that the chosen time-fractional derivative (Caputo or Riemann-Liouville) exactly recovers the memory kernel. This equivalence is load-bearing for all subsequent risk predictions.

Authors: We agree that an explicit verification strengthens the presentation. In the revised manuscript we will expand Section 3 with the full calculation: starting from the integro-differential generator of the asymmetric Lévy process, we apply the Fourier symbol to obtain the fractional Laplacian term for a general Lévy measure; we then show that the chosen Caputo time-fractional derivative exactly recovers the memory kernel convolution. These steps will be written out in detail so that the equivalence is self-contained and rigorous. revision: yes
Referee: [Numerical results and validation (Section 5)] The abstract and summary state that the PDE “accurately characterizes” long-term risk, yet the supplied text contains no error metrics, convergence rates, or direct Monte-Carlo comparisons that would quantify the discrepancy between solutions of the fractional PDE and the true safety/recovery probabilities of the original jump process. Without such verification, the generalization claims for the physics-informed learner cannot be assessed.

Authors: We accept the need for quantitative evidence. The revised Section 5 will include L² error norms between the physics-informed PDE solutions and independent Monte-Carlo estimates of the safety and recovery probabilities, convergence rates under spatial and temporal discretization, and side-by-side comparisons across multiple initial states and horizons. These metrics will directly support the accuracy and generalization statements. revision: yes

Circularity Check

0 steps flagged

No circularity: PDE derived from stochastic generator, not fitted or self-defined

full rationale

The paper states it derives the space-time fractional PDE directly from the infinitesimal generator of the Levy-jump process with memory kernel. No equations reduce the target safety/recovery probabilities to a fit on themselves, no self-citation is load-bearing for the generator-to-PDE step, and the physics-informed solver is applied after derivation to obtain numerical solutions rather than to define the PDE itself. The derivation chain therefore remains independent of the final risk quantities.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach rests on extensions of stochastic calculus to fractional operators for Levy processes and memory; no explicit free parameters or invented entities are named in the abstract, but fractional orders are implicitly required.

free parameters (1)

fractional orders (space and time)
The orders of the fractional derivatives must be chosen or fitted to match the jump and memory characteristics of the target system.

axioms (1)

domain assumption The underlying process is a stochastic differential equation with Levy jumps and long-range memory that admits a fractional PDE representation.
Invoked when stating that the derived PDE characterizes the true risk probabilities.

pith-pipeline@v0.9.0 · 5528 in / 1303 out tokens · 30845 ms · 2026-05-14T23:32:26.524208+00:00 · methodology

discussion (0)

Reference graph

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