We consider a simple random walk on $\mathbb{Z}^d$ started at the origin and stopped on its first exit time from $({-}L,L)^d \cap \mathbb{Z}^d$. Write L in the form $L = m N$ with $m = m(N)$ and N an integer going to infinity in such a way that $L^2 \sim A N^d$ for some real constant $A \gt 0$. Our main result is that for $d \ge 3$, the projection of the stopped trajectory to the N-torus locally converges, away from the origin, to an interlacement process at level $A d \sigma_1$, where $\sigma_1$ is the exit time of a Brownian motion from the unit cube $({-}1,1)^d$ that is independent of the interlacement process. The above problem is a variation on results of Windisch (2008) and Sznitman (2009).

We review and extend the class of hypergeometric Lévy processes explored in Kuznetsov and Pardo (2013) with a view to computing fluctuation identities related to stable processes. We give the Wiener-Hopf factorisation of a process in the extended class, characterise its exponential functional, and give three concrete examples arising from transformations of stable processes.

Consider N players, respectively owning x1, x2, …, xN monetary units, who play a sequence of games, winning from and losing to each other integer amounts according to fixed rules. The sequence stops as soon as (at least) one player is ruined. We are interested in the ruin process of these N players, i.e. in the probability that a given player is ruined first, and also in the expected ruin time. This problem is called the N-player ruin problem. In this paper, the problem is set up as a multivariate absorbing Markov chain with an absorbing state corresponding to the ruin of each player. This is then discussed in the context of phase-type distributions where each phase is represented by a vector of size N and the distribution has as many absorbing points as there are ruin events. We use this modified phase-type distribution to obtain an explicit solution to the N-player problem. We define a partition of the set of transient states into different levels, and on it give an extension of the folding algorithm (see Ye and Li (1994)). This provides an efficient computational procedure for calculating some of the key measures.

Motivated by edge behaviour reported for biological organisms, we show that random walks with a bias at a boundary lead to a discontinuous probability density across the boundary. We continue by studying more general diffusion processes with such a discontinuity across an interior boundary. We show how hitting probabilities, occupancy times and conditional occupancy times may be solved from problems that are adjoint to the original diffusion problem. We highlight our results with a biologically motivated example, where we analyze the movement behaviour of an individual in a network of habitat patches surrounded by dispersal habitat.

We consider a fluid queue with downward jumps, where the fluid flow rate and the downward jumps are controlled by a background Markov chain with a finite state space. We show that the stationary distribution of a buffer content has a matrix exponential form, and identify the exponent matrix. We derive these results using time-reversed arguments and the background state distribution at the hitting time concerning the corresponding fluid flow with upward jumps. This distribution was recently studied for a fluid queue with upward jumps under a stability condition. We give an alternative proof for this result using the rate conservation law. This proof not only simplifies the proof, but also explains an underlying Markov structure and enables us to study more complex cases such that the fluid flow has jumps subject to a nondecreasing Lévy process, a Brownian component, and countably many background states.

For a large class of neutral population models the asymptotics of the ancestral structure of a sample of n individuals (or genes) is studied, if the total population size becomes large. Under certain conditions and under a well-known time-scaling, which can be expressed in terms of the coalescence probabilities, weak convergence in DE([0,∞)) to the coalescent holds. Further the convergence behaviour of the jump chain of the ancestral process is studied. The results are used to approximate probabilities which are of certain interest in applications, for example hitting probabilities.