Well-posedness for Wright--Fisher SPDEs with measurable drift
Abstract
We prove weak existence and uniqueness in law of [0,1]-valued solutions for a class of scalar stochastic heat equations with measurable drifts b : [0,1] → ℝ, driven by Wright--Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities −Cu≤ b(u)≤ C(1−u) for some C<∞, compatible with the necessary condition b(1)≤ 0≤ b(0). These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.
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BibTeX
@misc{koskela2026wellposedness,
title = {{Well-posedness for Wright--Fisher SPDEs with measurable drift}},
author = {Jere Koskela and Oliver Tough},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.23518},
eprint = {2609.23518},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.23518},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Well-posedness for Wright--Fisher SPDEs with measurable drift
AU - Jere Koskela
AU - Oliver Tough
PY - 2026
DA - 2026/09/20
PB - arXiv
DO - 10.48550/arxiv.2609.23518
UR - https://arxiv.org/abs/2609.23518
AB - We prove weak existence and uniqueness in law of [0,1]-valued solutions for a class of scalar stochastic heat equations with measurable drifts b : [0,1] → ℝ, driven by Wright--Fisher noise. The only assumptions we require on the drift are Borel-measurability, and the one-way inequalities −Cu≤ b(u)≤ C(1−u) for some C<∞, compatible with the necessary condition b(1)≤ 0≤ b(0). These are vastly more general conditions than were previously available. Our proof relies on a stochastic duality between the solution of the stochastic heat equation, and a voting scheme running along the graph of a branching-coalescing particle system, generalising the class of drifts for which a dual process is available. Our results and their proof put the regularisation-by-noise result of Barnes, Mytnik and Sun, and its explanation, on a much more general footing.
ER -