# Minisymposium Mathematics cluster STAR

## Applied Probability

**Organizer & Chair: Onno Boxma (Technische Universiteit Eindhoven)**

Waaier 2, 14:15 - 16:15

The stochastics cluster STAR presents four lectures on various topics in applied probability. Keywords are Markov processes, Lévy processes, growth processes and random spatial structures.

**14:15-14:45 A scaling analysis of a cat and mouse Markov chain**

*Nelly Litvak (Universiteit Twente)*

Based on a Markov chain with discrete state space, a two-dimensional Markov chain is constructed. The first coordinate (the cat) behaves like the original Markov chain and the second component (the mouse) changes only when both coordinates are equal (i.e. when the mouse is found by the cat). We obtain an invariant measure of this Markov chain. We show that when the state space is infinite, and the initial Markov chain is positive recurrent and reversible, then the cat-and-mouse Markov chain is null recurrent. In this context, the scaling properties of the location of the second component, the mouse, are investigated in various situations: simple one- and two-dimensional random walks, and reflected random walks on non-negative integers. For several of these processes, a time scaling with rapid growth gives an interesting and surprising asymptotic behaviour related to limit results for occupation times and rare events of Markov processes. It is a joint work with Philippe Robert.

It is a joint work with Philippe Robert.

**14:45-15:15 A stochastic growth process where large finite clusters are frozen**

*Rob van den Berg (CWI and Vrije Universiteit Amsterdam)*

About twelve years ago, David Aldous, motivated by studies of sol-gel transitions, constructed a fascinating growth process for the binary tree. In this process clusters freeze as soon as they become infinite. It was pointed out by Itai Benjamini and Oded Schramm that such a process does not exist on (nice) two-dimensional grids. This motivated us to investigate the modified process, where clusters freeze as soon as they have diameter larger than or equal to a finite parameter N. In particular we were interested in the question whether this N-parameter model in two dimensions shows (compared with the process on trees) some form of ‘anomalous’ behaviour as N goes to ∞.

In this talk, which is based on cooperation with Demeter Kiss, Pierre Nolin and Bernardo de Lima, I will give a partial answer.

**15:15-15:45 On the number of collisions in lambda-coalescents**

*Alexander Gnedin (Universiteit Utrecht)*

The lambda-coalescent (introduced by Pitman and Sagitov) is a Markovian process in which particles merge according to certain rules, to eventually form a single cluster. We discuss the number of collisions (i.e. internal nodes of the coalescent tree) X for the process starting with n particles. The asymptotic behaviour of X for large n strongly depends on the concentration of the parameter measure near 0. In this talk we shall survey known limit theorems for X, with emphasis on the case when the coalescent can be coupled with an increasing Lévy process.

**15:45-16:15 A Lévy input fluid queue with workload and input regulation**

*Maria Vlasiou (Technische Universiteit Eindhoven)*

We consider a queuing model with the workload evolving between consecutive
i.i.d. exponential timers {e_{q}^{(i)}}_{
i=1,2,…} according to two spectrally positive Lévy
processes Y _{1} and Y _{2}, both reflected at 0. When the exponential clock e_{q}^{(i)} ends,
the additional state-dependent service requirement modifies the workload so that
the latter is equal to F_{i}^{(1)}(Y _{
k}(e_{q}^{(i)})) at epoch e_{
q}^{(1)} + + e_{
q}^{(i)} for some random
nonnegative i.i.d. functionals F_{i}^{(1)}. Moreover, at these moments the feedback
information is available and the label of the Laplace exponent is changed
according to the functional F^{(2)}(Y _{
k}(e_{q}^{(i)}), sup _{
s≤eq(i)}Y _{k}(s), inf _{s≤eq(i)}Y _{k}(s),k). For
example, F^{(2)}(x,y,z,k) = 1_{
{y≥K}} + 2 ⋆ 1_{{y<K}} if the workload process crossed a
fixed level K. In that case, we change the Lévy input process to Y _{1}, which
corresponds to the lighter load. Otherwise, if the load did not exceed the
threshold K we choose the process Y _{2}, which corresponds to the “regular load”.
We analyse the steady-state workload distribution for the general model and
provide several examples.