TL;DR
This survey reviews the historical and mathematical connections between Brownian motion, symmetrization, and complex analysis over the past 60 years, highlighting techniques for isoperimetric inequalities.
Contribution
It compiles and discusses key methods and results linking Brownian motion, symmetrization, and complex analysis, emphasizing isoperimetric inequalities and their proofs.
Findings
Connections between harmonic measure and exit probability
Techniques for isoperimetric inequalities in Brownian motion
Survey of mathematical methods in the field
Abstract
In this survey we explore the salient connections made between Brownian motion, symmetrization and complex analysis in the last 60 years starting with Kakutani's paper (1944) equating harmonic measure and exit probability. To exemplify these connections we will survey the techniques used in the literature to prove isoperimetric results for exit probabilities and Riesz capacities.
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Taxonomy
TopicsStochastic processes and financial applications · Financial Risk and Volatility Modeling · Stochastic processes and statistical mechanics