 Convergence to closed-form distribution for the backward SLEκ at some random times and the phase transition at κ=8Terry J. Lyons, Vlad Margarint and Sina Nejad Statistics & Probability Letters, 2024, vol. 205, issue C Abstract: We study a one-dimensional SDE that we obtain by performing a random time change of the backward Loewner dynamics in H. The stationary measure for this SDE has a closed-form expression. We show the convergence towards its stationary measure for this SDE, in the sense of random ergodic averages. The precise formula of the density of the stationary law gives a phase transition at the value κ=8 from integrability to non-integrability, that happens at the same value of κ as the change in behavior of the SLEκ trace from non-space filling to space-filling curve. Using convergence in total variation for the law of this diffusion towards stationarity, we identify families of random times on which the law of the arguments of points under the backward SLEκ flow converge to a closed form expression measure. For κ=4, this gives precise characterization for the random times on which the law of the arguments of points under the backward SLEκ flow converge to the uniform law. Keywords: Closed-form expression; Backward Loewner differential equation (search for similar items in EconPapers) Downloads: (external link) Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:eee:stapro:v:205:y:2024:i:c:s0167715223001827 Ordering information: This journal article can be ordered from DOI: 10.1016/j.spl.2023.109958 Access Statistics for this article Statistics & Probability Letters is currently edited by Somnath Datta and Hira L. Koul More articles in Statistics & Probability Letters from Elsevier |
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