{"ID":2870345,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2509.12921","arxiv_id":"2509.12921","title":"Non-parametric estimation of non-linear diffusion coefficient in parabolic SPDEs","abstract":"In this article, we introduce a novel non-parametric predictor, based on conditional expectation, for the unknown diffusion coefficient function $σ$ in the stochastic partial differential equation $Lu = σ(u)\\dot{W}$, where $L$ is a parabolic second order differential operator and $\\dot{W}$ is a suitable Gaussian noise. We prove consistency and derive an upper bound for the error in the $L^p$ norm, in terms of discretization and smoothening parameters $h$ and $\\varepsilon$. We illustrate the applicability of the approach and the role of the parameters with several interesting numerical examples.","short_abstract":"In this article, we introduce a novel non-parametric predictor, based on conditional expectation, for the unknown diffusion coefficient function $σ$ in the stochastic partial differential equation $Lu = σ(u)\\dot{W}$, where $L$ is a parabolic second order differential operator and $\\dot{W}$ is a suitable Gaussian noise....","url_abs":"https://arxiv.org/abs/2509.12921","url_pdf":"https://arxiv.org/pdf/2509.12921v1","authors":"[\"Martin Andersson\",\"Benny Avelin\",\"Valentin Garino\",\"Pauliina Ilmonen\",\"Lauri Viitasaari\"]","published":"2025-09-16T10:18:21Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[\"Diffusion Model\"]","has_code":false}