[Paper Review] SLE and alpha-SLE driven by Levy processes

Qing-Yang Guan, Matthias Winkel|ArXiv.org|Jun 27, 2006

Stochastic processes and statistical mechanics20 references3 citations

TL;DR

This paper introduces a generalization of Stochastic Loewner Evolution (SLE) driven by Lévy processes, specifically combining Brownian motion and symmetric $α$-stable Lévy processes. It establishes a phase transition at $κ=4$ for the Lebesgue measure of the cluster (zero or positive), and a further transition at $α=1$ linked to recurrence/transience of the driving process, with rigorous probabilistic proofs of hitting time behavior for complex plane points.

ABSTRACT

Stochastic Loewner Evolutions (SLE) with a multiple sqrt(kappa)B of Brownian motion B as driving process are random planar curves (if kappa<=4) or growing compact sets generated by a curve (if kappa>4). We consider here more general Levy processes as driving processes and obtain evolutions expected to look like random trees or compact sets generated by trees, respectively. We show that when the driving force is of the form sqrt(kappa)B+theta^(1/alpha)S for a symmetric alpha-stable Levy process S, the cluster has zero or positive Lebesgue measure according to whether kappa<=4 or kappa>4. We also give mathematical evidence that a further phase transition at alpha=1 is attributable to the recurrence/transience dychotomy of the driving Levy process. We introduce a new class of evolutions that we call alpha-SLE. They have alpha-self-similarity properties for alpha-stable Levy driving processes. We show the phase transition at a critical coefficient theta=theta_0(alpha) analogous to the kappa=4 phase transition.

Motivation & Objective

• To extend classical SLE, driven by Brownian motion, to more general Lévy processes as driving mechanisms.
• To investigate the geometric and probabilistic properties of the resulting random clusters in the complex upper half-plane.
• To determine under what conditions the cluster has zero or positive Lebesgue measure.
• To analyze the role of $α$-stability and recurrence/transience of the Lévy process in shaping the phase transitions of the growth process.
• To rigorously establish the hitting time behavior of points in $\overline{\mathbb{H}}\setminus\{0\}$ under different parameter regimes.

Proposed method

• Formalize SLE via the Loewner differential equation with a Lévy process $U_t = \sqrt{\kappa}B_t + \theta^{1/\alpha}S_t$ as driving function.
• Use conformal mapping techniques and the Loewner equation $\partial_t g_t(z) = 2/(g_t(z) - U_t)$ to describe cluster evolution.
• Apply probabilistic tools such as hitting time analysis, conditional expectations, and martingale arguments to study $\zeta(z) = \inf\{t \geq 0 : z \in K_t\}$.
• Introduce $\alpha$-SLE as a new class of evolutions with $\alpha$-self-similarity under $\alpha$-stable driving processes.
• Establish bounds on exit probabilities and conditional expectations using inequalities involving $|h_{1,\xi_n}|$ and $d_n = a_1^{-\beta} + n - 1$, leading to convergence results.
• Leverage recurrence/transience criteria for Lévy processes to interpret the phase transition at $\alpha=1$.

Experimental results

Research questions

• RQ1 How does the inclusion of a symmetric $\alpha$-stable Lévy process as a driving force affect the geometric structure of the SLE cluster?
• RQ2 What is the critical value of $\kappa$ at which the Lebesgue measure of the cluster transitions from zero to positive, and how does it depend on $\alpha$?
• RQ3 Does the recurrence or transience of the driving Lévy process induce a phase transition in the SLE evolution, and if so, at what value of $\alpha$?
• RQ4 Under what conditions is the hitting time $\zeta(z)$ almost surely finite for $z \in \overline{\mathbb{H}} \setminus \{0\}$?
• RQ5 Can the $\alpha$-SLE process be characterized by self-similarity properties under $\alpha$-stable driving processes?

Key findings

• For $0 \leq \kappa \leq 4$ and $U \not\equiv 0$, $\mathbb{P}(\zeta(z) = \infty) = 1$ for all $z \in \overline{\mathbb{H}} \setminus \{0\}$, meaning the cluster has zero Lebesgue measure.
• For $\kappa > 4$ and $1 \leq \alpha < 2$, $\mathbb{P}(\zeta(z) < \infty) = 1$ for all $z \in \overline{\mathbb{H}} \setminus \{0\}$, indicating the cluster has positive Lebesgue measure.
• For $\kappa > 4$ and $0 < \alpha < 1$, $0 < \mathbb{P}(\zeta(z) < \infty) < 1$ for all $z \in \overline{\mathbb{H}} \setminus \{0\}$, with $\lim_{z \to 0} \mathbb{P}(\zeta(z) < \infty) = 1$, showing intermediate behavior.
• The phase transition at $\alpha = 1$ corresponds to the recurrence/transience dichotomy of the $\alpha$-stable Lévy process, with isolated trees for $\alpha < 1$ and forests for $\alpha \geq 1$.
• The paper rigorously proves that the expected hitting time $\mathbb{E}_z[\zeta]$ is finite under the conditions of Theorem 1.4, using bounds on conditional probabilities and summability of $d_n^{-2}$ terms.
• The introduction of $\alpha$-SLE as a self-similar class of evolutions under $\alpha$-stable driving processes is validated through the existence of a critical coefficient $\theta_0(\alpha)$ analogous to the $\kappa=4$ transition.