'Probabilistic methods for solving fractional PDEs and generalized fractional PDEs' - Professor Vassili Kolokoltsov
We shall discuss recent achievements in the probabilistic interpretation and extensions
of the fractional PDEs of Caputo and Riemann-Liouville type and on the path integral representations for their solutions arising from this point of view. The main advantage of these path integral representations is their universality allowing to cover a variety of different problems in a concise unified way, and the possibility to yield solutions in a compact form that is explicitly stable with respect to the initial data and key parameters and is directly amenable to numeric schemes (Monte-Carlo simulation). The starting point for this development is the observation that from the point of view of stochastic analysis the Caputo and Riemann-Liouville derivatives
of order can be viewed as (regularized) generators of stable Lévy motions interrupted on crossing a boundary. This interpretation naturally suggests fully mixed, two-sided or even multidimensional generalizations of these derivatives, as well as a probabilistic approach to the analysis of the related equations. As application of these ideas we shall discuss wide classes of generalized fractional equations giving probabilistic interpretations of their solutions in terms of the Dynkin type martingales and/or chronological operator-valued extensions of the Feynman-Kac formulas.
'On Boltzmann-type equations in socio-economic applications' - Dr Marie-Therese Wolfram
Kinetic equations have become an indispensible tool to provide a quantitative description of diverse phenomena, such as semi-conductors, plasma physics, price and opinion formation, and more recently animal herding and pedestrian dynamics. In this talk we discuss how kinetic or so-called Boltzmann type models, can be used to describe the dynamics of many interacting particle systems using laws of statistical mechanics in pedestrian dynamics and opinion formation. Starting with the microscopic interaction laws, we derive the corresponding Boltzmann-type equations and discuss the behavior of solutions in different limits. Numerical simulations complete the picture of the rich dynamics and confirm the emergence of complex pattern, such as lane formation in bidirectional flows.
'Mathematical models meet (missing) patient data' - Professor Jane Hutton
As mathematicians, we enjoy the power of equations to express patterns and condense information. Statisticians consider how to connect equations with the world, collecting data and deciding on what is a good fit. Differential equations provide a plausible model for recovery from sprained ankle, but what data can we get from people? Stochastic process models for the effects of drugs on epileptic seizures make sense, but can people record when seizures occur? I will discuss some successes and failures in fitting.
'Poisson processes and Spatial Transportation Networks: network geodesics and Rayleigh random flights' - Professor Wilfrid Kendall
Scale-invariant random spatial networks (SIRSN) are remarkable random structures which provide patterns of random routes that are scale-invariant, thus modelling apparent scale-invariance in online maps. My talk will review the rather non-trivial theory establishing the existence of the Poisson line SIRSN and the known properties of network geodesics. In order to develop a good intuition about the behaviour of these geodesics, attention has turned to random scattering processes on the Poisson line SIRSN. This in turn leads to an axiomatization of abstract scattering processes (Markov chains which algebraically look like scattering processes), perhaps of wider interest in reliability theory. Ergodic theory (in particular a continuum version of the famous range theorem of Kesten, Spitzer and Whitman) can then be applied to produce insight into the behaviour of a randomly broken geodesic on the SIRSN.