Tangential Loewner hulls

Abstract

Through the Loewner equation, real-valued driving functions generate sets called Loewner hulls. We analyze driving functions that approach 0 at least as fast as a

Drivers, hitting times, and weldings in Loewner's equation

“…Lind [22] used connections between $$\xi, \tau$$ and $$\varphi$$ as part of her argument that a driver $$\xi$$ with Hölder‐1/2 seminorm $$|\xi|_{1/2}<4$$ generates a simple curve. She proved that, given $$|\xi|_{1/2}<4$$, $$\xi$$ welds exactly two points at each time, $$\tau (x) \asymp (x-\xi (0))^2$$, and $$\varphi$$ satisfies $$(\varphi (x-h)-\varphi (x))/(\varphi (x)-\varphi (x+h)) \asymp 1$$ [22, Lemma 3, Corollary 1, Lemma 4]. In our Lemma 3.1, we generalize the first result to hold whenever $$\xi$$ generates a simple curve (which is known to be a broader class than the $$\xi$$ satisfying $$|\xi|_{1/2}<4$$).…”

“…The literature on this question, in addition to the above-mentioned work of Lind [22] building off [26], appears to consist of [23,32] and [52].…”

“…Recall that well‐known elements of $$S$$ include linear drivers [15], drivers with one‐sided Hölder‐1/2 norm less than four [22, 23, 52] and, a.s., scaled Brownian motion $$\sqrt {\kappa}B(t)$$ when $$0 \leqslant \kappa \leqslant 4$$ [30].…”

“…whenever 𝑡 ⩽ 𝜏 + (𝑦 1 ; 𝜂), as follows from the growth (22) of intervals in the upward flow. Now, fix 𝑡 0 ∈ [0, 𝑇] and consider 𝑦 0 ∶= 𝜏 −1 + (𝑡 0 ; 𝜉) and ỹ0 ∶= 𝜏 −1 + (𝑡 0 ; ξ), and suppose, without loss of generality, that 𝑦 0 ⩽ ỹ0 .…”

“…This is a generalization of [3, Prop. 4.1$$(a)$$] and [22, Lemma 3]; the former, because this always holds, rather than only almost surely, and the latter, because we only assume $$\xi \in S([0,T])$$, not that $$\xi \in $$ Höl(1/2) with $$|\xi|_{1/2}<4$$. Lemma If $$\xi \in S([0,T])$$, $$0 < T \leqslant \infty$$, $$x \mapsto \tau (x) = \tau (x;\xi)$$ is continuous.…”