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Regularity of SLE in and refined GRR estimates.

Peter K Friz1, Huy Tran2, Yizheng Yuan2

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Summary

This study enhances understanding of Schramm-Loewner evolution (SLE) by proving joint Hölder continuity for SLE traces up to a higher parameter value. The research also establishes stochastic continuity for SLE traces, advancing the analysis of random fractal curves.

Keywords:
30C2060G1760G6060J6760K35

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Area of Science:

  • Stochastic processes
  • Conformal field theory
  • Geometric analysis

Background:

  • Schramm-Loewner evolution (SLE) is a fundamental tool for modeling random curves in 2D.
  • Previous work established almost sure Hölder continuity of the SLE trace for specific parameter ranges.
  • Understanding the regularity and continuity of SLE traces is crucial for their rigorous mathematical analysis.

Purpose of the Study:

  • To improve the known Hölder continuity results for SLE traces.
  • To establish stochastic continuity of the SLE trace for all parameter values.
  • To introduce a novel variation of the Garsia-Rodemich-Rumsey inequality for broader applications.

Main Methods:

  • Utilizing Loewner evolution with half-plane capacity parametrization.
  • Employing a novel variation of the Garsia-Rodemich-Rumsey inequality.
  • Analyzing the regularity and continuity properties of the resulting random field.

Main Results:

  • Achieved joint Hölder continuity of the SLE trace up to a higher parameter value (previously shown for a lower range).
  • Demonstrated stochastic continuity of the SLE trace as a continuous path for all parameter values.
  • Developed a new Garsia-Rodemich-Rumsey inequality with independent mathematical significance.

Conclusions:

  • The findings significantly advance the understanding of SLE trace regularity and continuity.
  • The improved continuity results have implications for the study of random fractal geometry.
  • The novel inequality provides a new analytical tool for related stochastic process research.