# P3110: Generic analytic Arnold diffusion in a priori stable systems

- ID: `P3110`
- Reference: `generic-analytic-arnold-diffusion-apriori-stable`
- Page: https://theoremdb.org/statements/P3110
- Record maturity: Reviewed problem with recorded work

## Problem

For a real-analytic steep integrable Hamiltonian \(h(I)\) with at least three degrees of freedom, is Arnold diffusion present for a generic sufficiently small real-analytic perturbation \(H_\varepsilon(I,\theta)=h(I)+\varepsilon f(I,\theta)\), in the sense that some orbit changes its action by an amount independent of \(\varepsilon\)?

### Context

Known frontier: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.

Open boundary: A general analytic genericity theorem for a priori stable steep systems remains open.

### Problem setup

- **Definition (a priori stable).** The unperturbed system is integrable without a built-in hyperbolic subsystem.
- **Definition (Arnold diffusion).** An orbit has order-one drift in action variables under an arbitrarily small perturbation.
- **Remark.** KAM tori occupy large measure and Nekhoroshev theory gives very long stability. In three or more degrees of freedom those tori do not separate the energy surface, leaving thin resonance channels where diffusion is expected.

### What counts as a solution

- State a standard Baire or prevalence notion of genericity and prove order-one diffusion for the corresponding analytic perturbations.
- Or disprove the generic claim by an open or residual obstruction class under the same topology.

## Status

OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open. [1](#reference-1) [2](#reference-2)

## Work

### Evidence for the current status

**Claim 1 (Current status and exact unresolved remainder).** OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.

The problem was checked as open on 2026-08-01.

The strongest neighboring result found in the cited sources is: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.

The exact unresolved remainder is: A general analytic genericity theorem for a priori stable steep systems remains open.

A complete resolution must meet the following acceptance conditions:
- State a standard Baire or prevalence notion of genericity and prove order-one diffusion for the corresponding analytic perturbations.
- Or disprove the generic claim by an open or residual obstruction class under the same topology.

### Background and intake notes

- Original intake status: OPEN as checked on 2026-08-01. Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.
- The release review checked 2 structured sources on 2026-08-01.
- Equivalent-formulation queries: generic analytic Arnold diffusion a priori stable open problem; Arnold diffusion analytic genericity stable Hamiltonian
- Strongest checked neighboring result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes.
- Exact unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open.

### Other known results

- **Claim 2** (supported): Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. [1](#reference-1) [2](#reference-2)

### Prior approaches

- **Route 1** (supported): The exact target, equivalent terminology, and 2025-2026 status evidence were checked on 2026-08-01. Strongest checked result: Generic diffusion is proved in broad a priori unstable systems and in several stable models or regularity classes. Unresolved remainder: A general analytic genericity theorem for a priori stable steep systems remains open. [1](#reference-1) [2](#reference-2)

### Open directions

- **Route 2** (reported): A general analytic genericity theorem for a priori stable steep systems remains open.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `generic-analytic-arnold-diffusion-apriori-stable`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>Patrick Bernard, “Arnold's Diffusion: from the a priori unstable to the a priori stable case”. arXiv:1203.2893 (2012). a priori stable discussion https://arxiv.org/abs/1203.2893
   - Also cited at P. Bernard, Arnold's diffusion: from the a priori unstable to the a priori stable case. a priori stable discussion
   - preprint; secondary source; arXiv:1203.2893, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Explains the transition to the hard stable case and outlines prospective mechanisms.
   - Source used to assess the problem's recorded status.
   - For Generic analytic Arnold diffusion in a priori stable systems: This is the dated publication status for the canonical target Generic analytic Arnold diffusion in a priori stable systems.
   - Source named by the research packet.
2. <a id="reference-2"></a>Qinbo Chen and Rafael de la Llave, “Analytic genericity of diffusing orbits in a priori unstable Hamiltonian systems”. Nonlinearity, vol.35, no.4, 2022. DOI 10.1088/1361-6544/ac50bb. arXiv:2103.03847 (2021). main genericity theorem https://arxiv.org/abs/2103.03847
   - preprint; primary source; arXiv:2103.03847, checked 2026-08-01; checked 2026-08-01
   - Source use: original_summary
   - Proves analytic genericity in a broad a priori unstable class, isolating the stable extension.
   - Source used to assess the problem's recorded status.
   - For Generic analytic Arnold diffusion in a priori stable systems: Proves analytic genericity in a broad a priori unstable class, isolating the stable extension.