There is wide interest in the oceanographic and engineering communities as to whether linear models are satisfactory for describing the largest and steepest waves in open ocean. This talk will give some background on the topic before describing some recent modelling. This concludes that non-linear physics produces only small increases in amplitude over that expected in a linear model — however, there are significant changes to the shape and structure of extreme wave-group caused by the non-linear physics.

Across much of the ablation region of the western Greenland Ice Sheet, hydro-fracture events related to supraglacial lake drainages rapidly deliver large volumes of meltwater to the bed of the ice sheet. We investigate what triggers the rapid drainage of a large supraglacial lake using a Network Inversion Filter (NIF) to invert a dense local network of GPS observations over three summers (2011-2013). The NIF is used to determine the spatiotemporal variability in ice sheet behavior (1) prior to lake drainage, and in response to (2) vertical hydro-fracture crack propagation and closure, (3) the opening of a horizontal cavity at the ice-sheet bed that accommodates the rapid injection of melt-water, and (4) extra basal slip due to enhanced lubrication. We find that the opening and propagation of each summer’s lake-draining hydro-fracture is preceded by a local stress perturbation associated with ice sheet uplift and enhanced slip above pre-drainage background velocities. We hypothesize that these precursors are associated with the introduction of meltwater to the bed through neighboring moulin systems.

Abstract:In the first part of the talk, using classical hypergeometric identities, I will compute the harmonic measure of finite intervals and their complementaries for the Lévy stable process on the line. This gives a simple and unified proof of several results by Blumenthal-Getoor-Ray, Rogozin, and Kyprianou-Pardo-Watson. In the second part of the talk, I will consider the two-dimensional Markov process based on the stable Lévy process and its area process. I will give two explicit formulae for the harmonic measure of the split complex plane. These formulae allow to compute the persistence exponent of the stable area process, solving a problem raised by Zhan Shi. This is based on two joint works with Christophe Profeta.

Abstract: It has been recently shown that rough volatility models reproduce very well the statistical properties of low frequency financial data. In such models, the volatility process is driven by a fractional Brownian motion with Hurst parameter of order 0.1. The goal of this talk is to explain how such fractional dynamics can be obtained from the behaviour of market participants at the microstructural scales.

Using limit theorems for Hawkes processes, we show that a rough volatility naturally arises in the presence of high frequency trading combined with metaorders splitting. This is joint work with Thibault Jaisson.