Driving functions and traces of the Loewner equation

Driving function of the vertical slit

“…We actually prove in Theorem 3.1 for a complete expression of λ(t) in terms of a series, and the constant C is also given explicitly. We note that the H ölder exponent of λ decreases from ∞ to 1/2, and remark that the case in [3] is for p = 1, the case in [15] is for p ∈ [1/3, ∞), and the case in [7], [10] and [16] corresponds to p = ∞ heuristically.…”

An introduction to the stochastic Loewner evolution

“…反之, 当给定连续函数 W (t) 时, 同样通过分析方程 (2.6) 有如下结论: 定理 2.8 [12] 给定连续函数 W (t) : (15) γ (14) γ (13) γ (12) γ (11) γ (10) γ (9) γ(8) γ (7) γ (6) γ (5) γ (4) γ(3)…”

“…反之不一定成立, 即给定连续函数 W (t), (2.1) 的解产生的紧致包不一定由曲线 生成. 关于这方面更深一步的研究参见文献[8][9][10].定义 2.2 给定连续函数 W (t) : [0, T ] → R, 考虑微分方程 (2.1) 的解所对应的紧致包 K t . 如果 存在一条曲线 γ : [0, T ] → H 使得对任意的 t, H t = H\K t 为 H\γ[0, t] 的无界连通分支, 则称 (2.1) 是 由 γ 生成的.…”

On tangential slit solution of the Loewner equation

“…The importance of the equation emerged again in the recent study of the stochastic Loewner evolution (SLE) due to Lawler, Schramm and Werner [9,10,11,19,8] and the references there, and Smirnov [20,21,22]. This also re-ignited the interest of the equation and its solution in the deterministic case [2,5,6,12,13,14,15,16,18,24,25].…”