Probability Surveys
- Probab. Surveys
- Volume 12 (2015), 55-103.
Conformal restriction and Brownian motion
Abstract
This survey paper is based on the lecture notes for the mini course in the summer school at Yau Mathematics Science Center, Tsinghua University, 2014.
We describe and characterize all random subsets $K$ of simply connected domain which satisfy the “conformal restriction” property. There are two different types of random sets: the chordal case and the radial case. In the chordal case, the random set $K$ in the upper half-plane $\mathbb{H}$ connects two fixed boundary points, say 0 and $\infty$, and given that $K$ stays in a simply connected open subset $H$ of $\mathbb{H}$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is any conformal map from $H$ onto $\mathbb{H}$ fixing 0 and $\infty $. In the radial case, the random set $K$ in the upper half-plane $\mathbb{H}$ connects one fixed boundary points, say 0, and one fixed interior point, say $i$, and given that $K$ stays in a simply connected open subset $H$ of $\mathbb{H}$, the conditional law of $\Phi(K)$ is identical to that of $K$, where $\Phi$ is the conformal map from $H$ onto $\mathbb{H}$ fixing 0 and $i$.
It turns out that the random set with conformal restriction property are closely related to the intersection exponents of Brownian motion. The construction of these random sets relies on Schramm Loewner Evolution with parameter $\kappa=8/3$ and Poisson point processes of Brownian excursions and Brownian loops.
Article information
Source
Probab. Surveys, Volume 12 (2015), 55-103.
Dates
Received: April 2015
First available in Project Euclid: 12 October 2015
Permanent link to this document
https://projecteuclid.org/euclid.ps/1444653628
Digital Object Identifier
doi:10.1214/15-PS259
Mathematical Reviews number (MathSciNet)
MR3411718
Zentralblatt MATH identifier
1339.60141
Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60J69
Keywords
Conformal invariance restriction property Brownian excursion Brownian loop Schramm Loewner evolution
Citation
Wu, Hao. Conformal restriction and Brownian motion. Probab. Surveys 12 (2015), 55--103. doi:10.1214/15-PS259. https://projecteuclid.org/euclid.ps/1444653628
