The Sharp Rate of Probabilistically Strong Convergence to the KPZ Equation
Abstract
One of the most common descriptions of solutions of singular SPDEs is their characterisation as the limit of solutions of renormalised smooth random PDEs. We quantify the speed of this convergence in the case of the KPZ equation. In particular, we show that the naïve guess that the rate is given by the distance of the noise regularity from the endpoint regularity for well-posedness is not correct. Instead, we obtain convergence at the larger rate 1/2 and show that this rate is sharp. This is achieved by considering the equation satisfied by the rescaled error, which is critical for variance blowup, and showing that this equation has a limit given by an affine linear singular SPDE driven by a new, independent noise. This result can be alternatively interpreted as identifying the asymptotic size and law of fluctuations of the solutions of the KPZ equation driven by mollified noise around their singular limit.
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BibTeX
@misc{gerencser2026sharp,
title = {{The Sharp Rate of Probabilistically Strong Convergence to the KPZ Equation}},
author = {Máté Gerencsér and Yueh-Sheng Hsu and Rhys Steele},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.02755},
eprint = {2609.02755},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.02755},
}
RIS
TY - UNPB
TI - The Sharp Rate of Probabilistically Strong Convergence to the KPZ Equation
AU - Máté Gerencsér
AU - Yueh-Sheng Hsu
AU - Rhys Steele
PY - 2026
DA - 2026/09/02
PB - arXiv
DO - 10.48550/arxiv.2609.02755
UR - https://arxiv.org/abs/2609.02755
AB - One of the most common descriptions of solutions of singular SPDEs is their characterisation as the limit of solutions of renormalised smooth random PDEs. We quantify the speed of this convergence in the case of the KPZ equation. In particular, we show that the naïve guess that the rate is given by the distance of the noise regularity from the endpoint regularity for well-posedness is not correct. Instead, we obtain convergence at the larger rate 1/2 and show that this rate is sharp. This is achieved by considering the equation satisfied by the rescaled error, which is critical for variance blowup, and showing that this equation has a limit given by an affine linear singular SPDE driven by a new, independent noise. This result can be alternatively interpreted as identifying the asymptotic size and law of fluctuations of the solutions of the KPZ equation driven by mollified noise around their singular limit.
ER -