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In this thesis I investigate various mathematical problems that are either connected to or loosely inspired by evolutionary biology. The main focus is on variants of existing models with a special emphasis on dynamic models or those in which the random quantities that define the model change over time. The first two chapters are concerned with accessibility percolation on tree-like graphs. I first look at a dynamic model on finite (but arbitrarily large) trees and then consider a static model on infinite (but locally finite) trees. For the dynamic model I consider two different scalings of the height of the tree, with the second scaling corresponding to a 'critical window' for the first scaling. For each of these scalings, I show that there exist times within an interval of polynomial size in the height of the tree, at which an accessible path exists (with probability converging to 1 as the height of the tree goes to infinity). For the infinite tree, a criterion is defined that quantifies the amount of branching in the tree, and this criterion is shown to determine a phase transition for the existence of an accessible path. In the second part of the thesis, I investigate variants of a model of adaptive dynamics, in which two mutants compete for fixation. The first variant involves an established type and an invading type, with the growth rate of the invading type subject to periodic switches between two values. I show that the length of time taken to invade is averaged provided the period length grows sufficiently slowly with respect to the carrying capacity K. If the period length decreases to zero with increasing K, the probability of invasion is itself averaged. The second variant involves two non-established types with one type initiated after a delay. The probability that either type reaches a size proportional to K is calculated for different initial starting sizes and it is shown that for fixed starting sizes, there is a phase transition in the duration of the allowed delay on the log K timescale.

The aim of the thesis is to characterise new ways of path-conditioning of a d-dimensional isotropic stable Lévy process and consider their time-reversed paths. In our analysis, we use both of classical potential theory approach and recently-developed methods around the theory of self- similar Markov processes. In doing so, we have the opportunity to consider the role of certain harmonic/excessive functions that have not been previously studied. In the first part of the thesis, Chapter 3, we consider an oscillatory conditioned attraction of the stable Lévy process to a subset of the unit sphere or a hyperplane. We characterise the hitting distribution as well as the time-reversed process of this conditioned processes in the sense of Hunt-Nagasawa duality for Markov processes. The resulting time-reversed processes have the same distribution as the unconditioned stable Lévy process itself when issued from the corresponding subset of the unit sphere or an hyperplane. In the second part, Chapter 4, we condition the stable Lévy process to remain either outside or inside of the sphere and we characterise the same conditioned attraction to a subset of the unit sphere. The methods of the previous chapter are not applicable in this case. We use instead recent developments in the representation of d-dimensional isotropic stable Lévy processes as a self-similar Markov processes. In particular, we use a characterisation of the point of closest/furthest reach from the unit sphere for the stable Lévy process. As in the first part, we characterise the hitting distribution as well as the time-reversed process of the newly conditioned processes via Hunt-Nagasawa duality. The resulting time-reversed processes have the law of stable Lévy processes conditioned to stay away from the unit sphere and the process conditioned to stay inside the unit sphere and continuously absorbed at the origin correspondingly. We also extend the conditioned processes as well as its time-reversed processes to be issued from the boundary of the domain in which they live. Finally, we would like point out that the methodology that we use in the Chapter 4 was only possible for a subset of the unit sphere and we could not extend to the setting of the hyperplane. The reason for this is that we use the law of the point of closest reach from the unit sphere to define the conditioning. There is no such result for the case of an hyperplane so far. Thus, for further work, we aim to characterise one-sided attraction to a subset of an hyperplane.

The largest part of this thesis is concerned with the study of a fragmentation process in which rectangles break up into progressively smaller pieces at rates that depend on their shape. Long, thin rectangles are more likely to break quickly, and are also more likely to split along their longest side. We are interested in the evolution of the system at large times: how many fragments are there of different shapes and sizes, and how did they reach that state? We give an almost sure growth rate along paths by studying an equivalent branching random walk. Our analysis is highly technical due to the spatial dependence of the rates and the fact that we work under weaker assumptions than the usual large deviations regime for random walks. In the second part of the thesis we focus on a different, but related problem: estimating the probability that the paths of a random walk stay close to a given function. We prove a small deviation result about the unscaled paths of either a compound Poisson process, or a random walk in discrete time. Our proof strategy involves a Brownian motion approximation on smaller time intervals, which allows us to take advantage of the sharpest estimates currently available on the probability that a Brownian motion lies in a tube about a given function.