Second-order fields for stochastic partial differential equations
Abstract
In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form (∂t − Δ) Φε = −P(Φε) + ξε, where P is a polynomial of degree greater than or equal to $2$, ξε is the white-noise after being convoluted (in space) by the heat kernel Kε = eε Δ. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits Φ= limε → 0Φε, we characterise the limit of Φerr=limε → 0 ε−1(Φε−Φ) as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato--Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.
Related
- Mean field singular stochastic partial differential equations
- The Neumann problem for stochastic partial differential equations
- A Class of Stochastic Partial Differential Equations with Jumps in Fluid Dynamics: Large and Moderate Deviation Asymptotics
- Backward stochastic partial differential equations with unbounded random coefficients
- Diffusion-Based Stochastic Operator Networks for Uncertainty Quantification in Stochastic Partial Differential Equations
BibTeX
@misc{barros2026secondorder,
title = {{Second-order fields for stochastic partial differential equations}},
author = {Eldon Barros and Leandro Chiarini and Milton Jara},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09540},
eprint = {2609.09540},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09540},
}
RIS
TY - UNPB
TI - Second-order fields for stochastic partial differential equations
AU - Eldon Barros
AU - Leandro Chiarini
AU - Milton Jara
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09540
UR - https://arxiv.org/abs/2609.09540
AB - In this article, we study the second-order fluctuation of the solutions of one-dimensional polynomial stochastic partial differential equations (SPDEs) of the form (∂_t − Δ) Φ_ε = −P(Φ_ε) + ξ_ε, where P is a polynomial of degree greater than or equal to $2$, ξ_ε is the white-noise after being convoluted (in space) by the heat kernel K_ε = e^ε Δ. More precisely, taking advantage of the local solutions of pointwise well-posedness of limits Φ= lim_ε → 0Φ_ε, we characterise the limit of Φ^err=lim_ε → 0 ε^−1(Φ_ε−Φ) as the solution of a more irregular stochastic partial differential equation. This is performed by applying the Da Prato--Debussche decomposition in the non-linear equation and characterising the limit of each of the terms. We also discuss possible characterisations of second-order fluctuations performed for higher-dimensional SPDEs in the weakly-coupled regime.
ER -