Splitting algorithm and normed convergence for drawing the random fractal Loewner curves

2021/10/20 by Jiaming Chen, Chen, Jiaming, Vlad Margarint +1
Economics, Econometrics and Finance · Mathematics · #30 Functions of a complex variable #60 Probability theory and stochastic processes #FOS: Mathematics #Probability (math.PR) #Random Matrices and Applications #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2110.10631

openalex publication_date 2021/10/20 · openalex created_date 2021/10/25 · openalex updated_date 2026/07/28

Abstract

In the first part of the paper we propose and study the approximation of the SLE_κ trace via the Ninomiya-Victoir splitting algorithm. We prove the uniform convergence in probability with respect to the sup-norm to the distance between the SLE_κ trace and the output of the Ninomiya-Victoir splitting algorithm when applied in the context of the Loewner differential equation. Further investigations on the L^p-norm convergence is also exhibited, shedding light on the more delicate convergence structure. In the second part we show the uniform convergence of the approximation of the SLE_κ trace obtained using a different scheme that is based on the linear interpolation of the Brownian driving force.

Citations

• Contour lines of the two-dimensional discrete Gaussian free field

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