Department of Mathematics, Imperial College London
Doi
Abstract
This thesis is a collection of collaborative research work which uses field-theoretic techniques to
approach three different areas of stochastic dynamics: Branching Processes, First-passage times
of processes with are subject to both white and coloured noise, and numerical and analytical
aspects of first-passage times in fractional Brownian Motion.
Chapter 1 (joint work with Rosalba Garcia Millan, Johannes Pausch, and Gunnar Pruessner,
appeared in Phys. Rev. E 98 (6):062107) contains an analysis of non-spatial branching processes
with arbitrary offspring distribution. Here our focus lies on the statistics of the number of
particles in the system at any given time. We calculate a host of observables using Doi-Peliti
field theory and find that close to criticality these observables no longer depend on the details
of the offspring distribution, and are thus universal.
In Chapter 2 (joint work with Ignacio Bordeu, Saoirse Amarteifio, Rosalba Garcia Millan,
Nanxin Wei, and Gunnar Pruessner, appeared in Sci. Rep. 9:15590) we study the number of
sites visited by a branching random walk on general graphs. To do so, we introduce a fieldtheoretic
tracing mechanism which keeps track of all already visited sites. We find the scaling
laws of the moments of the distribution near the critical point.
Chapter 3 (joint work with Gunnar Pruessner and Guillaume Salbreux, submitted, arXiv:
2006.00116) provides an analysis of the first-passage time problem for stochastic processes
subject to white and coloured noise. By way of a perturbation theory, I give a systematic and
controlled expansion of the moment generating function of first-passage times.
In Chapter 4, we revise the tracing mechanism found earlier and use it to characterise three
different extreme values, first-passage times, running maxima, and mean volume explored. By
formulating these in field-theoretic language, we are able to derive new results for a class of
non-Markovian stochastic processes.
Chapter 5 and 6 are concerned with the first-passage time distribution of fractional Brownian
Motion. Chapter 5 (joint work with Kay Wiese, appeared in Phys. Rev. E 101 (4):043312)
introduces a new algorithm to sample them efficiently. Chapter 6 (joint work with Maxence
Arutkin and Kay Wiese, submitted, arXiv:1908.10801) gives a field-theoretically obtained perturbative
result of the first-passage time distribution in the presence of linear and non-linear
drift.Open Acces
