Chemical diffusion master equations: analytical solutions and applications

This thesis is the compilation of 3 of mine and my supervisor’s research, that are present in Chapters 4, 5 and 6 from the corresponding papers (Lanconelli et al., 2023, Lanconelli and Perçin, 2024a) and (Lanconelli and Perçin, 2024b) respectively. Our work is centered around the model, which is called the Chemical Diffusion Master Equation (CDME), which enables us to assign probabilities to different configurations of particles undergoing in a reaction of interest at time t. Due to this nature of the model, the model fits into the category of stochastic hybrid systems, where continuous dynamics and discrete events coexist in the same process, which makes the analysis more realistic and more complex compared to other well known models such as Chemical Master Equation (CME) (see (Erban and Chapman, 2020, del Razo et al., 2022)). Because the model is very complex most of the time the research including CDME is only focused at computer simulations. However in Chapters 4 and 5 we report a general method to treat the system analytically for any reaction where the reaction and diffusion are considered to be independent. This is very important given the low abundance of such solutions. Moreover we believe seeing an explicit solution gives better insights on how the parameters of the model plays a role. Moreover as will be seen in Chapter 6, the CDME of a reaction-diffusion system where the reaction and diffusion are not independent will be reported and solved explicitly with an approach particular to the process considered.