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We consider radial Loewner evolution driven by unimodular Lévy processes. We rescale the hulls of the evolution by capacity, and prove that the weak limit of the rescaled hulls exists. We then study a random growth model obtained by driving the Loewner equation with a compound Poisson process. The process involves two real parameters: the intensity of the underlying Poisson process and a localization parameter of the Poisson kernel which determines the jumps. A particular choice of parameters yields a growth process similar to the Hastings-Levitov HL(0) model. We describe the asymptotic behavior of the hulls with respect to the parameters, showing that growth tends to become localized as the jump parameter increases. We obtain deterministic evolutions in one limiting case, and Loewner evolution driven by a unimodular Cauchy process in another. We show that the Hausdorff dimension of the limiting rescaled hulls is equal to 1. Using a different type of compound Poisson process, where the Poisson kernel is replaced by the heat kernel, as driving function, we recover one case of the aforementioned model and SLE(κ) as limits.1991 Mathematics Subject Classification. Primary: 30C35, 60D05; Secondary: 60K35.

The observational method in geotechnical engineering is an acceptable verification method for limit states in Eurocode 7, but the method is rarely used despite its potential savings. Some reasons may be its unclear safety definition and the lack of guidelines on how to establish whether the observational method is more favourable than conventional design. In this paper, we challenge these issues by introducing a reliability con­straint on the observational method and propose a probabilistic optimi­sation methodology that aids the decision-making engineer in choosing between the observational method and conventional design. The method­ology suggests an optimal design after comparing the expected utilities of the considered design options. The methodology is illustrated with a practical example, in which a geotechnical engineer evaluates whether the observational method may be favourable in the design of a rock pillar. We conclude that the methodology may prove to be a valuable tool for deci­sion-making engineers’ everyday work with managing risks in geotech­nical projects.

QC 20160923

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