The volume of catchment discharge that reaches a stream via the overland flow path is critical for water quality prediction, because it is via this pathway that most particulate pollutants are generated and transported to the stream channel, via surface erosion processes. When it rains, spatial variation in the soil infiltration rate leads to the formation and reabsorption of rivulets on the surface, and local topography determines the coalescence of rivulets.

We consider the question of how coalescence facilitates overland flow using a highly abstracted version of the problem, in which the drainage pattern is represented by a Galton-Watson tree. We show that as the rate of rainfall increases there is a distinct phase-change: when there is no stream coalescence the critical point occurs when the rainfall rate equals the infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the infiltration rate, and increasing the amount of coalescence increases the total expected runoff.

The Hastings-Levitov planar aggregation models describe growing random clusters on the complex plane, built by iterated composition of random conformal maps. A striking feature of these models is that they can be used to define natural off–lattice analogues of several fundamental discrete models, such as the Eden model or Diffusion Limited Aggregation, by tuning the correlation between the defining maps appropriately. In this talk I will discuss shape theorems and fluctuations of large clusters in the weak correlation regime.

Joint work with James Norris and Amanda Turner.

Given a finite partially ordered set $(P,\prec)$, a linear extension of it is a total order on the set which is compatible with the original partial order. A comparable pair of elements is two elements $x,y\in P$ for which we have $x\prec y$ or $y\prec x$. It is reasonable to ask about the extent of any relationship between the number of comparable pairs and the number of linear extensions, with an initial rough intuition being that fewer comparable pairs should correspond to more linear extensions. We will focus in this talk on the case of dense posets where a strictly positive proportion of pairs are comparable, though other cases will be considered as well. I shall focus on probabilistic aspects in this version of the talk, including the original motivating problem of estimating how many linear extensions a random interval order (which has roughly two-thirds of pairs comparable) possesses.

This talk is based on joint work with Vasileios Iliopoulos and Colin McDiarmid.

In a seminal work from the 50s of the last century, William Feller classified all one-dimensional diffusions on $-\infty \leq a {}<{} b\leq \infty$ in terms of their ability to access the boundary (Feller's test for explosions) and to enter the interior from the boundary. Feller's technique is restricted to diffusion processes as the corresponding differential generators allow the use of Hille-Yosida theory. In the present article we study exit and entrance from $\mathbb{R}$ for the most natural generalization: jump diffusions \begin{align*} dZ_t=\sigma(Z_{t-})\,dX_t \end{align*} driven by stable L\'evy processes for $\alpha\in (0,2)$. We derive necessary and sufficient conditions on $\sigma$ so that (i) non-exploding solutions exist and (ii) the corresponding transition semigroup extends to an entrance point at `infinity'. We show that the presence of jumps allows different types of infinity corresponding to entrance from $+\infty$, $-\infty$ and simultaneous entrance from both.

Our methodology pulls together recent results on the entrance behavior and space-time transformations for self-similar Markov processes, recent as well as classical facts and fluctuation identities for (killed) stable processes, some recent results on perpetual integrals for L\'evy processes and classical duality theory for general Markov processes. Arguments for $\alpha>1$ and $\alpha= 1$ differ significantly in the details by virtue of non-existence of local times for $\alpha=1$.