Guillaume Baverez
We prove that the welding homeomomorphism of SLE$_4$ is almost surely $\log$-regular. In a previous version of this work, we had erroneously deduced its removability from this property. Nevertheless, the $\log$-regularity does provide some information and could lead to future developments.
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Regularity of the SLE$_4$ uniformizing map and the SLE$_8$ trace
研究SLE_4统一化映射的连续模，证明其为( log δ^{-1} )^{-1/3 + o(1)}，并展示SLE_4不满足Jones-Smirnov条件。同时证明SLE_8的连续模为( log δ^{-1} )^{-1/4 + o(1)}，验证Alvisio和Lawler的猜想。
We revisit regularity of SLE trace, for all $κ\neq 8$, and establish Besov regularity under the usual half-space capacity parametrization. With an embedding theorem of Garsia--Rodemich--Rumsey type, we obtain finite moments (and hence almost surely) optimal variation regularity with index $\min (1 + κ/ 8, 2) $, improving on previous works of Werness, and also (optimal) Hölder regularity à la Johansson Viklund and Lawler.
This paper has been withdrawn by the author due to essential mistakes in some previous versions.
Schramm-Loewner evolution (SLE$_κ$) is classically studied via Loewner evolution with half-plane capacity parametrization, driven by $\sqrtκ$ times Brownian motion. This yields a (half-plane) valued random field $γ= γ(t, κ; ω)$. (Hölder) regularity of in $γ(\cdot,κ;ω$), a.k.a. SLE trace, has been considered by many authors, starting with Rohde-Schramm (2005). Subsequently, Johansson Viklund, Rohde, and Wong (2014) showed a.s. Hölder continuity of this random field for $κ< 8(2-\sqrt{3})$. In this paper, we improve their result to joint Hölder continuity up to $κ< 8/3$. Moreover, we show that th…
When studying stochastic processes, it is often fruitful to have an understanding of several different notions of regularity. One such notion is the optimal Hölder exponent obtainable under reparametrization. In this paper, we show that the chordal SLE_kappa path in the unit disk for kappa less than or equal to 4 can be reparametrized to be Hölder continuous of any order up to 1/(1+kappa/8). From this result, we obtain that the Young integral is well defined along such SLE paths with probability one, and hence that SLE admits a path-wise notion of integration. This allows for us to consider th…
Refined regularity of SLE
本文证明了容量参数化下SLE轨迹的精细变差与Hölder型正则性，简化SLE₈相关证明步骤并得到对应模，还对κ≥8的一致化映射证明了时间一致的正则性估计，结果通过正向Loewner微分方程分析得出。
Miller and Sheffield introduced the notion of an imaginary surface as an equivalence class of pairs of simply connected proper subdomains of $\mathbb{C}$ and Gaussian free fields (GFFs) on them under the conformal equivalence. They considered the situation in which the conformal maps are given by a chordal Schramm--Loewner evolution (SLE). In the present paper, we construct GFF-valued processes on $\mathbb{H}$ (the upper half-plane) and $\mathbb{O}$ (the first orthant of $\mathbb{C}$) by coupling a GFF with a multiple SLE evolving in time on each domain. We prove that a GFF on $\mathbb{H}$ and…
This paper examines how close the chordal $\SLE_κ$ curve gets to the real line asymptotically far away from its starting point. In particular, when $κ\in(0,4)$, it is shown that if $β>β_κ:=1/(8/κ-2)$, then the intersection of the $\SLE_κ$ curve with the graph of the function $y=x/(\log x)^β$, $x>e$, is a.s. bounded, while it is a.s. unbounded if $β=β_κ$. The critical $\SLE_4$ curve a.s. intersects the graph of $y=x^{-(\log\log x)^α}$, $x>e^e$, in an unbounded set if $α\le 1$, but not if $α>1$. Under a very mild regularity assumption on the function $y(x)$, we give a necessary and sufficient in…
Regularity of the Schramm-Loewner evolution: Up-to-constant variation and modulus of continuity
本文研究了SLE的正则性，确定了至常数变化和连续模的最优界，并证明了自然参数化由ψ-变化的细网格极限给出。
Singular paths spaces and applications
本文研究零点处带量化奇异性的路径与粗糙路径，建立奇异Hölder空间及相应积分理论，并用于解释SLE边界效应、统一粗糙波动率奇异核及归约重整化问题。
An Exact Hausdorff Measure as Natural Length for SLE
该研究针对弦SLE_κ，证明其精确豪斯多夫测度是Lawler与Rezaei构造的占据测度的确定性正常数倍，明确了SLE的自然长度相关测度性质。
Quasi-invariance for SLE welding measures
本文研究SLE焊接测度的性质，证明其在Weil-Petersson圆周同胚的群作用下具有拟不变性，为理解SLE焊接的分布规律提供了新的理论结果。
A note on regularity and separation for the stochastic Allen-Cahn equation with logarithmic potential
本文针对带对数势的随机Allen-Cahn方程证明其精细时空正则性，并由此建立解的轨迹与势垒严格分离的随机分离性质，将已有结果从p大于空间维数的p-Laplace算子情形推广到经典对数势情形。
The Wigner-Dyson-Gaudin-Mehta conjecture asserts that the local eigenvalue statistics of large real and complex Hermitian matrices with independent, identically distributed entries are universal in a sense that they depend only on the symmetry class of the matrix and otherwise are independent of the details of the distribution. We present the recent solution to this half-century old conjecture. We explain how stochastic tools, such as the Dyson Brownian motion, and PDE ideas, such as De Giorgi-Nash-Moser regularity theory, were combined in the solution. We also show related results for log-gas…
Intrinsic regularity in the discrete log-Sobolev inequality
本文研究了离散对数Sobolev不等式中极值子的正则性，证明其对数Lipschitz常数受$\log d$的通用倍数限制，并推导出有限状态空间上可逆Markov链的对数Sobolev常数下界，确立了稀疏参数$\log d$作为从Markov扩散转移到离散链的通用代价。
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