Random Structures, Applied Probability and Computation
LMS Research School on Probability at the University of Liverpool
Plenary Talks
Invariant distributions of Lévy-type processes and related questions - Anita Behme
Abstract:
We establish a distributional equation as a criterion for invariant measures of Markov processes associated to Lévy-type operators. It is obtained via a characterization of infinitesimally invariant measures of the associated generators. Particular focus is put on the one-dimensional case where the distributional equation becomes a Volterra-Fredholm integral equation, and on Lévy-type processes solving stochastic differential equations. The results are compared to other methods of determining invariant measures, applied on several examples and an outlook on possible applications in Monte Carlo simulations is presented.
Global freezing - Igor Kortchemski
Abstract:
This is joint work with Etienne Bellin and Arthur Blanc-Renaudie, Emmanuel Kammerer.
It's hard to kill fake news - Remco van der Hofstad
Abstract:
Empirical findings have shown that many real-world networks are scale-free, in the sense that there is a high variability in the number of connections of the elements of the networks. Spurred by these empirical findings, models have been proposed for such networks. In this talk, we investigate the spread of fake news on them.
We assume that news starts spreading from a source using a first-passage percolation rumour spread dynamics. The source later realises that the news is in fact wrong. After this realisation, it starts spreading the correct news. We make the (optimistic) assumption that a vertex, once having heard the correct version of the news item, will only spread the correct information. As such, we are modelling misinformation rather than fake news, with fake news being able to sustain on a network even longer.
Our results show that in many settings, even when the correct news spreads faster, the incorrect news is likely to reach a large part of the network. We distinguish between the incorrect news weakly surviving, meaning that it reaching a growing number of vertices, and strong survival, where the incorrect news reaches a positive proportion of the vertices. We give explicit criteria for the incorrect news to weakly and strongly survive on the configuration model, which is one of the most popular networks models.
This lecture is based on joint work with Seva Shneer, and builds on earlier work with Gerard Hooghiemstra and Shankar Bhamidi.
Brownian motion with asymptotically normal reflection in unbounded domains: from transience to stability - Aleksandar Mijatovic
Abstract:
In this talk we quantify the asymptotic behaviour of multidimensional drifltess diffusions in domains unbounded in a single direction, with asymptotically normal reflections from the boundary. We identify the critical growth/contraction rates of the domain that separate stability, null recurrence and transience. In the stable case we prove existence and uniqueness of the invariant distribution and establish the polynomial rate of decay of its tail. We also establish matching polynomial upper and lower bounds on the rate of convergence to stationarity in total variation. All exponents are explicit in the model parameters that determine the asymptotics of the growth rate of the domain, the interior covariance, and the reflection vector field. Proofs are probabilistic, and use upper and lower tail bounds for additive functionals up to return times to compact sets, for which we develop novel sub/supermartingale criteria, applicable to general continuous semimartingales. Time permitting, I will discuss the main ideas behind the proofs in the talk. This is joint work with Miha Bresar (Warwick) and Andrew Wade (Durham).
Aleksandar's YouTube Channel: HERE
It contains some short videos explaining the ideas about Aleksandar's recent work.