2017/02/26 by Zhan, Dapeng · 4 citations
#30C #60G #FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.1702.08026
We use Minkowski content (i.e., natural parametrization) of SLE to construct several types of SLEκ loop measures for κ∈(0,8). First, we construct rooted SLEκ loop measures in the Riemann sphere \widehat\mathbb C, which satisfy Möbius covariance, conformal Markov property, reversibility, and space-time homogeneity, when the loop is parametrized by its (1+\frac κ8)-dimensional Minkowski content. Second, by integrating rooted SLEκ loop measures, we construct the unrooted SLEκ loop measure in \widehat\mathbb C, which satisfies Möbius invariance and reversibility. Third, we extend the SLEκ loop measures from \widehat\mathbb C to subdomains of \widehat\mathbb C and to two types of Riemann surfaces using Brownian loop measures, and obtain conformal invariance or covariance of these measures. Finally, using a similar approach, we construct SLEκ bubble measures in simply/multiply connected domains rooted at a boundary point. The SLEκ loop measures for κ∈(0,4] give examples of Malliavin-Kontsevich-Suhov loop measures for all c≤ 1. The space-time homogeneity of rooted SLEκ loop measures in \widehat\mathbb C answers a question raised by Greg Lawler.