Splitting algorithm and normed convergence for drawing the random fractal Loewner curves
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arXiv
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| Format: | Preprint |
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2021
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| _version_ | 1866909289510076416 |
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| author | Chen, Jiaming Margarint, Vlad |
| author_facet | Chen, Jiaming Margarint, Vlad |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2110_10631 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Splitting algorithm and normed convergence for drawing the random fractal Loewner curves Chen, Jiaming Margarint, Vlad Probability 60 Probability theory and stochastic processes, 30 Functions of a complex variable 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. |
| title | Splitting algorithm and normed convergence for drawing the random fractal Loewner curves |
| topic | Probability 60 Probability theory and stochastic processes, 30 Functions of a complex variable |
| url | https://arxiv.org/abs/2110.10631 |