Abstract
Spatially dependent birth–death processes can be modelled by kinetic models such as the BBGKY hierarchy. Diffusion in infinite dimensional systems can be modelled with Brownian motion in Hilbert space. In this work Doi field theoretic formalism is utilised to establish dualities between these classes of processes. This enables path integral methods to calculate expectations of duality functions. These are exemplified with models ranging from stochastic cable signalling to jump-diffusion processes.
Highlights
Birth–death processes are concerned with fluctuations in the size of a population of interest, such as growing populations of cells, chemical reactions between molecules, and connectivity of networks, for example
If birth–death processes are generalised to spatially dependent systems, it is natural to enquire, firstly, how the field theoretic approaches to deriving dualities in [42, 44] can be adapted, and secondly, how do the results compare to those found via Martingale techniques in [14, 15]
These are the questions considered in this work, where we develop path integral approaches to analyse duality between spatial birth–death models and infinite dimensional diffusion processes
Summary
Birth–death processes are concerned with fluctuations in the size of a population of interest, such as growing populations of cells, chemical reactions between molecules, and connectivity of networks, for example. Standard approaches either have no spatial component and are just concerned with population size, or assume the spatially dependent population is fully mixed, with position playing no crucial role. Chemical reaction fronts exhibit non-homogeneous spatial behaviour, and incorporating spatial effects into the birth–death interactions is important. The theory behind stochastic analysis of fluctuating populations can be traced back to the master equation, originally developed by Kolmogorov [37]. Approaches to birth–death processes underwent crucial development by Kendall [34] and Karlin and McGregor [32, 33].
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