Paper · Mar 30, 2020

Minimum curvature flow and martingale exit times

Martin Larsson, Johannes Ruf

math.APWider flowsarXiv:2003.13611v2
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Abstract

We study the following question: What is the largest deterministic amount of time that a suitably normalized martingale can be kept inside a convex body in ? We show, in a viscosity framework, that equals the time it takes for the relative boundary of to reach as it undergoes a geometric flow that we call (positive) minimum curvature flow. This result has close links to the literature on stochastic and game representations of geometric flows. Moreover, the minimum curvature flow can be viewed as an arrival time version of the Ambrosio–Soner codimension- mean curvature flow of the -skeleton of . Our results are obtained by a mix of probabilistic and analytic methods.

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Submitted
Mar 30, 2020
Last revised
Jan 11, 2022 (v2)
Subjects
math.AP, math.PR
Mean curvature flowOther curvature flows

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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 2016, 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 Ancient solutions of geometric flows with curvature pinching (Susanna Risa and Carlo Sinestrari, 2017)4Risa 2017 Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow (Vicent Gimeno and Vicente Palmer, 2018)1Gimeno 2018 Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs (Nils Dabrock, Martina Hofmanová and Matthias Röger, 2019)3Dabrock 2019 The Lₚ Gauss dual Minkowski problem (Na Fu and Jianping Sun, 2024)6Fu 2024 A game approach to free boundary problems of anisotropic forced mean curvature flow equations (Takuya Sato, 2025)8Sato 2025 A two-player zero-sum probabilistic game that approximates the mean curvature flow (Irene Gonzalvez, Alfredo Miranda, Julio D. Rossi et al., 2025)7Gonzalvez 2025 Existence of martingale solutions for a stochastic weighted mean curvature flow of graphs (Qi Yan and Xiang-Dong Li, 2025)2Yan 2025 Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space (Xinxiu Zhang, 2026)5Zhang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Parabolicity, Brownian escape rate and properness of self-similar solutions of the direct and inverse Mean Curvature Flow

    Vicent Gimeno and Vicente PalmerWider flows

    Topics Other curvature flows · Shared terms “Exit Time”, “following”

  2. Existence of martingale solutions for a stochastic weighted mean curvature flow of graphs

    Qi Yan and Xiang-Dong LiWider flows

    Topics Mean curvature flow · Shared terms “martingale”, “stochastic”

  3. Existence of martingale solutions and large-time behavior for a stochastic mean curvature flow of graphs

    Nils Dabrock, Martina Hofmanová and Matthias RögerWider flows

    Topics Mean curvature flow · Shared terms “martingale”, “stochastic”, “deterministic”

  4. Ancient solutions of geometric flows with curvature pinching

    Susanna Risa and Carlo SinestrariWider flows

    Topics Other curvature flows · Shared terms “geometric flows”

  5. Modified support measures and Alexandrov-Fenchel type inequalities for nonsmooth h-convex bodies in hyperbolic space

    Xinxiu ZhangWider flows

    Topics Other curvature flows · Shared terms “convex body”

  6. The Gauss dual Minkowski problem

    Na Fu and Jianping SunWider flows

    Topics Other curvature flows · Shared terms “convex body”

  7. A two-player zero-sum probabilistic game that approximates the mean curvature flow

    Irene Gonzalvez, Alfredo Miranda, Julio D. Rossi et al.Wider flows

    Topics Mean curvature flow · Shared terms “probabilistic”, “deterministic”

  8. A game approach to free boundary problems of anisotropic forced mean curvature flow equations

    Takuya SatoWider flows

    Topics Mean curvature flow · Shared terms “game representation”, “deterministic”

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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974)

In the guidebook

  1. 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.