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Stochastic Loewner evolution in doubly connected domains. (English) Zbl 1054.60104

Stochastic Loewner evolution (SLE) is a random growth process of plane sets in simply connected domains. The evolution is described by the Loewner differential equation with the one-dimensional Brownian motion as the driving term. The paper attempts a generalization of O. Schramm’s theory [Isr. J. Math. 118, 221–288 (2000; Zbl 0968.60093)] to multiply connected domains and focuses of the doubly connected case. Two families of radial and annulus SLE are introduced and their equivalence is demonstrated. Links of the annulus SLE with the scaling limit of the loop-erased conditional random walk are established.
The disc SLE, which describes a random process of growing compact subsets of a simply connected domain, has been obtained as the limit case of the annulus SLE. In particular, depending on the specific value of the Loewner parameter \(\kappa \), we have: for \(\kappa =6\) one has a locality property and the same law as the hull generated by a plane Brownian motion stopped on hitting the boundary, while for \(\kappa =2\) there follows that the disc SLE is the reversal of radial SLE stated from a random point on the boundary with harmonic measure.

MSC:

60K35 Interacting random processes; statistical mechanics type models; percolation theory
58J65 Diffusion processes and stochastic analysis on manifolds
35R60 PDEs with randomness, stochastic partial differential equations
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics
30C35 General theory of conformal mappings

Citations:

Zbl 0968.60093
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