Weak-Form Recovery of Stochastic Generators and Dynamical Invariants
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914570918952960 |
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| author | A, Eshwar R Honnavar, Gajanan V. |
| author_facet | A, Eshwar R Honnavar, Gajanan V. |
| contents | Spectral gaps, Kramers escape rates, and position-dependent relaxation timescales are dynamical invariants encoded in the infinitesimal generator $\Lop$ of a stochastic flow. We show that weak projection of the governing Itô SDE onto temporal test functions produces an endogeneity bias of order $O(T\,\dt^{3/2})$ that grows with the observation window and cannot be eliminated by additional data. Projecting instead onto spatial Gaussian kernels removes the bias exactly: $\mathcal{F}_{t_n}$-measurability and the tower property guarantee unbiased regression rows at every step. The resulting framework jointly identifies the drift $b(x)$ and diffusion $a(x)$ from a single sparse regression, producing an explicit symbolic enerator amenable to spectral analysis. Validation on three benchmark systems yields coefficient errors below 5%, stationary-density total-variation distances below 0.01, and autocorrelation functions that faithfully reproduce true relaxation timescales. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_20904 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Weak-Form Recovery of Stochastic Generators and Dynamical Invariants A, Eshwar R Honnavar, Gajanan V. Methodology Mathematical Physics Dynamical Systems Chaotic Dynamics Data Analysis, Statistics and Probability Machine Learning 60H10, 62M09, 65C30, 93E12 Spectral gaps, Kramers escape rates, and position-dependent relaxation timescales are dynamical invariants encoded in the infinitesimal generator $\Lop$ of a stochastic flow. We show that weak projection of the governing Itô SDE onto temporal test functions produces an endogeneity bias of order $O(T\,\dt^{3/2})$ that grows with the observation window and cannot be eliminated by additional data. Projecting instead onto spatial Gaussian kernels removes the bias exactly: $\mathcal{F}_{t_n}$-measurability and the tower property guarantee unbiased regression rows at every step. The resulting framework jointly identifies the drift $b(x)$ and diffusion $a(x)$ from a single sparse regression, producing an explicit symbolic enerator amenable to spectral analysis. Validation on three benchmark systems yields coefficient errors below 5%, stationary-density total-variation distances below 0.01, and autocorrelation functions that faithfully reproduce true relaxation timescales. |
| title | Weak-Form Recovery of Stochastic Generators and Dynamical Invariants |
| topic | Methodology Mathematical Physics Dynamical Systems Chaotic Dynamics Data Analysis, Statistics and Probability Machine Learning 60H10, 62M09, 65C30, 93E12 |
| url | https://arxiv.org/abs/2603.20904 |