Pathwise Global-in-Time Existence for the generalised KPZ Equation in the Full Subcritical Regime
Abstract
We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely L∞ initial data, rather than the more usual setting of Cθ initial data with θ> 0. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard (1+1)-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.
Related
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BibTeX
@misc{sauer2026pathwise,
title = {{Pathwise Global-in-Time Existence for the generalised KPZ Equation in the Full Subcritical Regime}},
author = {Jonas Sauer and Rhys Steele},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.09096},
eprint = {2609.09096},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.09096},
}
RIS
TY - UNPB
TI - Pathwise Global-in-Time Existence for the generalised KPZ Equation in the Full Subcritical Regime
AU - Jonas Sauer
AU - Rhys Steele
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.09096
UR - https://arxiv.org/abs/2609.09096
AB - We provide a pathwise proof of global-in-time well-posedness for the generalised KPZ equation in the full subcritical regime by an adaptation of the strategy recently applied to the generalised Parabolic Anderson Model in [ES26]. Since this strategy relies crucially on the assumption that control of the supremum norm is sufficient to continue the solution, the main additional ingredient required is a treatment of the initial layer for gKPZ with merely L^∞ initial data, rather than the more usual setting of C^θ initial data with θ> 0. Our approach is based on an expansion around a deterministic profile followed by the introduction of an integrating factor in order to remove the terms which have critical scaling at time $0$. For pedagogical purposes, we first demonstrate the proof techniques in the case of the standard (1+1)-dimensional KPZ equation before turning to the more computationally involved case of generalised KPZ.
ER -