I am interested in systems of interacting one-dimensional Brownian particles, which observe instantaneous annihilation or coalescence. Building on my MSc dissertation, I am currently considering models with annihilation and immigration. The approach taken is to embed the interacting particle system in the Brownian Web (BW) and utilise duality to derive formulae which may be exploited, for example, to obtain the n-point densities.
The BW is a system of one-dimensional coalescing Brownian particles where, informally, particles are started at all points in time and space. Introduced by Arratia, it was further explored by Tóth and Werner. In particular, there exists a dual system of (backwards in time) coalescing particles and crucially, but perhaps counter-intuitively, there is an almost sure non-crossing property of the forwards and backwards paths. One way to aid intuition is to consider the discrete analogue, a system of coalescing random walks. Here the dual system with non-crossing property can easily be constructed (see here, Appendix). A simple colouring argument reveals an annihilating system within the BW.
By embedding the system of interacting Brownian particles in the BW, one can use duality and the non-crossing property to translate questions about forwards particles into questions about backwards particles. For example, in a coalescent non-immigration model, the probability that forwards particles nucleated at time zero do not end up in certain intervals at a given time is equal to the probability that the open set, formed by backwards particles started at the interval end points, does not contain any of these nucleation points. By similar considerations one may derive formulae for models with immigration, annihilation and for multiple intervals. Differentiating and taking limits appropriately in such formulae gives the n-point densities.
Working with Roger Tribe and Oleg Zaboronski, my MSc dissertation concerns an interacting Brownian particle system with pairwise immigration. The model is constructed as a limit of continuous processes. For various initial conditions, the fixed-time n-point densities are shown to be given by Pfaffian point processes.
Outside of maths, my passion is music. I have a large music collection and am a self-taught pianist. My accumulation of instruments includes an 1870’s harmonium (Mason & Hamlin), for the folk days, and a 1970’s monophonic synthesizer (Roland SH-2000), for the funk.