Tightness results for infinite-slit limits of the chordal Loewner equation. (English) Zbl 1390.60303
Summary: In this note, we consider a multi-slit Loewner equation with constant coefficients that describes the growth of multiple SLE curves connecting \(N\) points on \(\mathbb {R}\) to infinity within the upper half-plane. For every \(N\in \mathbb {N}\), this equation yields a measure-valued process \(t\mapsto \{\alpha _{N,t}\},\) and we are interested in the limit behaviour as \(N\rightarrow \infty .\) We prove tightness of the sequence \(\{\alpha _{N,t}\}_{N\in \mathbb {N}}\) under certain assumptions and address some further problems. Moreover, we investigate a similar situation in which all slits are trajectories of a certain quadratic differential.
MSC:
| 60J67 | Stochastic (Schramm-)Loewner evolution (SLE) |
| 37L05 | General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations |
Keywords:
chordal Loewner equation; stochastic Loewner evolution; multiple SLE; complex Burgers equation; tightness; quadratic differentialsReferences:
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