Loewner deformations driven by the Weierstrass function

Abstract

The Loewner differential equation provides a way of encoding growing families of sets into continuous real-valued functions. Most famously, Schramm-Loewner evolution (SLE) consists of the growing random families of sets that are encoded via the Loewner equation by a multiple of Brownian motion. The purpose of this paper is to study the families of sets encoded by a multiple of the Weierstrass function, which is a deterministic analog of Brownian motion. We prove that there is a phase transition in this setting, just as there is in the SLE setting. 159 4. Proof of the phase transition 163 Acknowledgement 164 References 164 MSC2010: 30C35.

Effect of random time changes on Loewner hulls

“…Since this function is continuous but nowhere differentiable, it serves as a deterministic analogue of Brownian motion. Similar to SLE, the Loewner hulls driven by c W , studied in [17], have a phase transition. In particular for c small enough c W generates simple curves, but for c large enough the hulls are not simple.…”

“…In particular for c small enough c W generates simple curves, but for c large enough the hulls are not simple. In [17], it is also shown that near a local maximum W t grows faster than an appropriately scaled square root curve. Therefore, Theorem 4.1(b) implies that c W (E t ) does not generate a simple curve.…”

“…To prove this, we use a result in [17] to first derive general criteria (Theorem 4.1) for verifying whether the curves generated by time-changed functions are simple or non-simple. The proof of Theorem 1.1 also relies on a deep relationship between Brownian motion and a 3-dimensional Bessel process given in [32] and local behaviors of Bessel processes studied in [31].…”

Tangential Loewner hulls

“…The first statment can be established by a short computation (see, for instance, [LRob,Lemma 2.3]). The second statement follows from a comparison between the driving functions λ and k √ 1 − t (see, for instance, [LRob,proof of Theorem 1.1]).…”

“…The first statment can be established by a short computation (see, for instance, [LRob,Lemma 2.3]). The second statement follows from a comparison between the driving functions λ and k √ 1 − t (see, for instance, [LRob,proof of Theorem 1.1]). While Lemma 2.1 can be used to determine when a Loewner hull is not a simple curve, the following theorem gives a large class of hulls that are simple curves.…”