{"ID":2898130,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.04035","arxiv_id":"2507.04035","title":"Divergence-Kernel method for scores of random systems","abstract":"We derive the divergence-kernel formula for the scores of random dynamical systems, then formally pass to the continuous-time limit of SDEs. Our formula works for multiplicative noise systems over any period of time; it does not require hyperbolicity. We also consider several special cases: (1) for additive noise, we give a pure kernel formula; (2) for short-time, we give a pure divergence formula; (3) we give a formula which does not involve scores of the initial distribution. Based on the new formula, we derive a pathwise Monte-Carlo algorithm for scores, and demonstrate it on the 40-dimensional Lorenz 96 system with multiplicative noise.","short_abstract":"We derive the divergence-kernel formula for the scores of random dynamical systems, then formally pass to the continuous-time limit of SDEs. Our formula works for multiplicative noise systems over any period of time; it does not require hyperbolicity. We also consider several special cases: (1) for additive noise, we g...","url_abs":"https://arxiv.org/abs/2507.04035","url_pdf":"https://arxiv.org/pdf/2507.04035v1","authors":"[\"Angxiu Ni\"]","published":"2025-07-05T13:20:24Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.DS\",\"math.NA\",\"math.OC\"]","methods":"[]","has_code":false}