Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity
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| Format: | Preprint |
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2022
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| _version_ | 1866912841337929728 |
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| author | Holden, Nina Yuan, Yizheng |
| author_facet | Holden, Nina Yuan, Yizheng |
| contents | We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrisation. More precisely, denoting by $d\in(0,2]$ the dimension of the curve, we show the following.
1. The optimal $ψ$-variation is $ψ(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in the sense that $η$ is a.s. of finite $ψ$-variation for this $ψ$ and not for any function decaying more slowly as $x \downarrow 0$.
2. The optimal modulus of continuity is $ω(s) = c\,s^{1/d}(\log s^{-1})^{1-1/d}$, i.e. for some random $c>0$ we have $|η(t)-η(s)| \le ω(t-s)$ a.s., while this does not hold for any function $ω$ decaying faster as $s \downarrow 0$.
3. $\limsup_{t\downarrow 0} |η(t)|\,\big(t^{1/d}(\log\log t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in $(0,\infty)$.
We also show that the natural parametrisation of SLE is given by the fine mesh limit of the $ψ$-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2211_15609 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity Holden, Nina Yuan, Yizheng Probability Complex Variables 60J67, 60G17 (Primary), 60G18, 30C20 (Secondary) We find optimal (up to constant) bounds for the following measures for the regularity of the Schramm-Loewner evolution (SLE): variation regularity, modulus of continuity, and law of the iterated logarithm. For the latter two we consider the SLE with its natural parametrisation. More precisely, denoting by $d\in(0,2]$ the dimension of the curve, we show the following. 1. The optimal $ψ$-variation is $ψ(x)=x^d(\log\log x^{-1})^{-(d-1)}$ in the sense that $η$ is a.s. of finite $ψ$-variation for this $ψ$ and not for any function decaying more slowly as $x \downarrow 0$. 2. The optimal modulus of continuity is $ω(s) = c\,s^{1/d}(\log s^{-1})^{1-1/d}$, i.e. for some random $c>0$ we have $|η(t)-η(s)| \le ω(t-s)$ a.s., while this does not hold for any function $ω$ decaying faster as $s \downarrow 0$. 3. $\limsup_{t\downarrow 0} |η(t)|\,\big(t^{1/d}(\log\log t^{-1})^{1-1/d}\big)^{-1}$ is a.s. equal to a deterministic constant in $(0,\infty)$. We also show that the natural parametrisation of SLE is given by the fine mesh limit of the $ψ$-variation. As part of our proof, we show that every stochastic process whose increments satisfy a particular moment condition attains a certain variation regularity. |
| title | Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity |
| topic | Probability Complex Variables 60J67, 60G17 (Primary), 60G18, 30C20 (Secondary) |
| url | https://arxiv.org/abs/2211.15609 |