An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Adrian Martini; Avi Mayorcas

doi:10.1007/s40072-024-00343-y

An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Adrian Martini, Avi Mayorcas

University of Oxford

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Abstract

Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

Original language	English
Pages (from-to)	956-1033
Number of pages	78
Journal	Stochastics and Partial Differential Equations: Analysis and Computations
Volume	13
Issue number	2
Early online date	24 Jan 2025
DOIs	https://doi.org/10.1007/s40072-024-00343-y
Publication status	Published - Jun 2025

Acknowledgements

We would like to express our gratitude to A. Etheridge, B. Fehrman, N. Perkowski and W. van Zuijlen for helpful discussions during the writing of this manuscript. We also wish to thank S. Mahdisoltani for bringing the subject of linear fluctuating hydrodynamics to our attention, in particular through the paper [40].

Funding

A. Martini was supported by the Engineering and Physical Sciences Research Council Doctoral Training Partnerships [Grant Number EP/R513295/1] and by the Lamb & Flag Scholarship of St John\u2019s College, Oxford. Part of this work was completed during A. Martini\u2019s participation in the Junior Trimester Program \u2018Stochastic modelling in the life science: From evolution to medicine\u2019 at the Hausdorff Research Institute for Mathematics, funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany\u2019s Excellence Strategy\u2014EXC-2047/1\u2014390685813. Work on this paper was undertaken during A. Mayorcas\u2019s tenure as INI-Simons Post Doctoral Research Fellow hosted by the Isaac Newton Institute for Mathematical Sciences (INI) participating in programme Frontiers in Kinetic Theory, and by the Department of Pure Mathematics and Mathematical Statistics (DPMMS) at the University of Cambridge. This author would like to thank INI and DPMMS for support and hospitality during this fellowship, which was supported by Simons Foundation (award ID 316017) and by Engineering and Physical Sciences Research Council (EPSRC) Grant Number EP/R014604/1.

Funders	Funder number
Lamb & Flag Scholarship of St John’s College, Oxford
Hausdorff Research Institute for Mathematics
Engineering and Physical Sciences Research Council	EP/R513295/1
Deutsche Forschungsgemeinschaft (DFG)	EXC-2047/1—390685813
Simons Foundation	EP/R014604/1, 316017

Keywords

Dean–Kawasaki equation
Linear fluctuating hydrodynamics
Parabolic-elliptic Keller–Segel model
Paracontrolled distributions
Singular stochastic partial differential equation

ASJC Scopus subject areas

Statistics and Probability
Modelling and Simulation
Applied Mathematics

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10.1007/s40072-024-00343-yLicence: CC BY

Cite this

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title = "An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions",

abstract = "Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green{\textquoteright}s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.",

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AB - Using the method of paracontrolled distributions, we show the local well-posedness of an additive-noise approximation to the fluctuating hydrodynamics of the Keller–Segel model on the two-dimensional torus. Our approximation is a non-linear, non-local, parabolic-elliptic stochastic PDE with an irregular, heterogeneous space-time noise. As a consequence of the irregularity and heterogeneity, solutions to this equation must be renormalised by a sequence of diverging fields. Using the symmetry of the elliptic Green’s function, which appears in our non-local term, we establish that the renormalisation diverges at most logarithmically, an improvement over the linear divergence one would expect by power counting. Similar cancellations also serve to reduce the number of diverging counterterms.

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An additive-noise approximation to Keller–Segel–Dean–Kawasaki dynamics: local well-posedness of paracontrolled solutions

Abstract

Acknowledgements

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Keywords

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