Paper · Jun 10, 2024

Exploring Stochastic Mean Curvature Flow on Networks Using Ito Calculus

Roman Bahadursingh

math.DGWider flowsarXiv:2406.16920
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As on arXiv

Abstract

In this paper, we investigate the stochastic mean curvature flow (SMCF) on networks, a niche area within stochastic processes and geometric analysis. By applying Ito calculus, we analyze the evolution of network structures influenced by random perturbations. We derive a stochastic differential equation (SDE) for the network edges and utilize numerical simulations to study the stability, long-term behavior, and pattern formation in these systems. Our results offer new insights into the dynamics of complex networks under stochastic influences and open pathways for future research in stochastic geometry.

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Submitted
Jun 10, 2024
Subjects
math.DG, math.ST
DOI
10.13140/RG.2.2.21595.37923/1
Mean curvature flowStability & uniquenessNumerics & visualization

Its neighbourhood

Connected papers

The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.

Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II (David Garfinkle, James Isenberg, Dan Knopf et al., 2022)4Garfinkle 2022 Stability of piecewise flat Ricci flow (Rory Conboye, 2023)1Conboye 2023 Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow (David Garfinkle, James Isenberg, Dan Knopf et al., 2024)2Garfinkle 2024 Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group (Giovanna Citti, Nicolas Dirr, Federica Dragoni et al., 2024)6Citti 2024 Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds (Miroslav Kolar and Daniel Sevcovic, 2025)7Kolar 2025 A curvature flow approach to dorsal closure modelling (Shuhui He, Ben Whale, Glen Wheeler et al., 2025)5He 2025 Ricci Flow on Weighted Digraphs with Balancing Factor (Shuliang Bai, Rui Li, Shuang Liu et al., 2025)8Bai 2025 Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow (Jihui Wang, Matteo Lulli and Antonino Marciano, 2026)3Wang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Stability of piecewise flat Ricci flow

    Rory Conboye

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”

  2. Asymptotic behavior of unstable perturbations of the Fubini-Study metric in Ricci flow

    David Garfinkle, James Isenberg, Dan Knopf et al.

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”, “perturbation”

  3. Noise-induced stabilization of Schwarzschild–AdS black holes under stochastic Ricci flow

    Jihui Wang, Matteo Lulli and Antonino Marciano

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “stochastic”, “simulations”, “perturbation”

  4. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

    David Garfinkle, James Isenberg, Dan Knopf et al.Wider flows

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical simulations”, “perturbation”

  5. A curvature flow approach to dorsal closure modelling

    Shuhui He, Ben Whale, Glen Wheeler et al.Wider flows

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “simulations”, “numerical”

  6. Horizontal mean curvature flow as a scaling limit of a mean field equation in the Heisenberg group

    Giovanna Citti, Nicolas Dirr, Federica Dragoni et al.Wider flows

    Topics Numerics & visualization

  7. Mean Curvature Flow of Closed Curves Evolving in Two Dimensional Manifolds

    Miroslav Kolar and Daniel SevcovicWider flows

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical”

  8. Ricci Flow on Weighted Digraphs with Balancing Factor

    Shuliang Bai, Rui Li, Shuang Liu et al.

    Topics Numerics & visualization, Stability & uniqueness · Shared terms “numerical”

Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005) Numerics & visualization Computing and visualising Ricci flow: simulations of neckpinches and critical behaviour, pictures of rotationally symmetric flows, and… Start with: Deforming metrics in the direction of their Ricci tensors (DeTurck, 1983)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.