Directed mean curvature flow in noisy environment

Andris Gerasimovičs; Martin Hairer; Konstantin Matetski

doi:10.1002/cpa.22158

Directed mean curvature flow in noisy environment

Andris Gerasimovičs
, Martin Hairer
, Konstantin Matetski

Imperial College London
Michigan State University

Research output: Contribution to journal › Article › peer-review

2 Citations (SciVal)

Abstract

We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

Original language	English
Pages (from-to)	1850-1939
Number of pages	90
Journal	Communications on Pure and Applied Mathematics
Volume	77
Issue number	3
Early online date	3 Oct 2023
DOIs	https://doi.org/10.1002/cpa.22158
Publication status	Published - 31 Mar 2024
Externally published	Yes

Bibliographical note

Funding Information:
The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2 , as well as Francesco Pedullá for pointing out a mistake in an earlier version of Lemma 7.12 . AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS‐1953859 (transferred to DMS‐2321493).

Funding

The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2, as well as Francesco Pedullá for pointing out a mistake in an earlier version of Lemma 7.12. AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS-1953859 (transferred to DMS-2321493). The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2 , as well as Francesco Pedullá for pointing out a mistake in an earlier version of Lemma 7.12 . AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS‐1953859 (transferred to DMS‐2321493).

Funders	Funder number
National Science Foundation	DMS-2321493, DMS‐1953859
The Leverhulme Trust
Royal Society

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

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Cite this

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title = "Directed mean curvature flow in noisy environment",

abstract = "We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.",

author = "Andris Gerasimovi{\v c}s and Martin Hairer and Konstantin Matetski",

note = "Funding Information: The authors would like to thank Ajay Chandra for numerous useful discussions of the renormalisation and coherence in Section 3.2 , as well as Francesco Pedull{\'a} for pointing out a mistake in an earlier version of Lemma 7.12 . AG gratefully acknowledges the financial support by the Leverhulme Trust through Hendrik Weber's Philip Leverhulme Prize. MH was supported by the Royal Society through a research professorship. KM was partially supported by NSF grant DMS‐1953859 (transferred to DMS‐2321493). ",

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N2 - We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

AB - We consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite-dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

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Directed mean curvature flow in noisy environment

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