Particle processes in which particles are assigned a mass and randomly move around in space but also undergo random splitting are called branching processes. At a splitting event additional particles are produced, either dividing the mass of the parent among them or introducing new mass, all in a random way. More generally, one may think of a random tree-like structure in space and time. Random objects that fit this description (considering them forwards or backwards in time) include branching Markov processes, continuous-state branching processes, superprocesses, coalescent and (growth) fragmentation processes, Fleming-Viot processes and continuous random trees. One is interested in both the propogation of mass as well as the evolution of population density.
Integrable probability is a research area that focuses on the study of probability models that arise in integrable systems, such as random matrix theory, interacting particle systems, and stochastic partial differential equations. The main goal is to understand the behavior of these models and derive exact solutions using techniques from integrable systems theory, such as the Bethe ansatz and Riemann-Hilbert methods. Integrable probability has applications in physics, statistics, and computer science.
This field studies large systems of particles and uses statistical mechanics to investigate how particle interactions affect the system's behavior. We also research phase transitions and fundamental laws that govern atomic and molecular behavior.
In recent years, there has been a focus on Markov processes with path discontinuities. There is a rich variety of processes we are interested in, all of which, at their heart, are based around an underlying Poisson point process on different structural spaces. These include Lévy processes, self-similar Markov processes and Markov additive processes. Of particular interest are path decompositions that decompose the path of the process of interest into smaller correlated constituent parts, thus allowing a characterisation of large and small-scale behaviour, both in space and time, which ultimately explains how it explores space with time.
Particular interests lie in exploring hypoelliptic diffusion processes, which are subject to underlying constraints but still move over all parts of the phase space, and at the interface of stochastic analysis with differential geometry where the study of random processes on manifolds reveals links between geometric features and stochastic aspects.
Stochastic partial differential equations (SPDEs) are mathematical models used to describe physical phenomena that exhibit random behavior over time and space. In particular, critical equations are at the heart of the research at Warwick. These are SPDEs where the balance between the nonlinear and linear terms in the equation is delicate, leading to critical phenomena such as power-law behavior and scaling limits. The study of SPDEs is an active area of research in mathematics and has applications in physics, biology and finance.