Regularity of SLE in and refined GRR estimates

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.