Past

A self-interacting random walk is a random process evolving in an environment depending on its past behaviour.

The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis [2] on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.

Although introduced by similar definitions, these processes show some significantly different behaviours, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as perturbations, and sometimes changing the nature of this dynamical system.

The purpose of our talk is to present our recent results on the subject [1,3,4,6].

Bibliography

[1] M. Bena

Numerous physical systems are justifiably modelled as Markov processes. However,

in practical applications the (usually implicit) assumptions concerning accurate

measurement of the system are often a fair departure from what is possible in

reality. In general, this lack of exact information is liable to render the

In Dept of Statistics

Recently random planar structures, such as planar graphs and outerplanar graphs, have received much attention. Typical questions one would ask about them are the following: how many of them are there, can we sample a random instance uniformly at random, and what properties does a random planar structure have ? To answer these questions we decompose the planar structures along their connectivity. For the asymptotic enumeration we interpret the decomposition in terms of generating funtions and derive the asymptotic number, using singularity analysis. For the exact enumeration and the uniform generation we use the so-called recursive method: We derive recursive counting formulas along the decomposition, which yields a deterministic polynomial time algorithm to sample a planar structure that is uniformly distributed. In this talk we show how to apply these methods to several labeled planar structures, e.g., planar graphs, cubic planar graphs, and outerplanar graphs.

In Dept of Statistics

/notices/events/abstracts/applied-analysis/tt05/Heywood.pdf

A firm issues a convertible bond. At each subsequent time, the bondholder

must decide whether to continue to hold the bond, thereby collecting coupons, or

to convert it to stock. The bondholder wishes to choose a conversion strategy to

maximize the bond value. Subject to some restrictions, the bond can be called by

the issuing firm, which presumably acts to maximize the equity value of the firm

by minimizing the bond value. This creates a two-person game. We show that if

the coupon rate is below the interest rate times the call price, then conversion

should precede call. On the other hand, if the dividend rate times the call

price is below the coupon rate, call should precede conversion. In either case,

the game reduces to a problem of optimal stopping. This is joint work with Mihai

Sirbu.

A scale-invariant moving finite element method is proposed for the

adaptive solution of nonlinear partial differential equations. The mesh

movement is based on a finite element discretisation of a scale-invariant

conservation principle incorporating a monitor function, while the time

discretisation of the resulting system of ordinary differential equations

may be carried out using a scale-invariant time-stepping. The accuracy and

reliability of the algorithm is tested against exact self-similar

solutions, where available, and a state-of-the-art $h$-refinement scheme

for a range of second and fourth order problems with moving boundaries.

The monitor functions used are the dependent variable and a monitor

related to the surface area of the solution manifold.

In Somerville

We review the analytic transformations allowing to construct standard bridges from a semistable Markov process, with indec 1/2, enjoying the time inversion property. These are generalized and some of there properties are studied. The new family maps the space of continuous real-valued functions into a family which is the topic of our focus. We establish a simple and explicit formula relating the distributions of the first hitting times of each of these by the considered semi-stable process

When two single server queues have the same arrivals process, this is said to be a `fork-join queue'. In the case where the arrivals and service processes are Brownian motions, the queue lengths process is a reflecting Brownian motion in the nonnegative orthant. Tan and Knessl [1996] have given a simple explicit formula for the stationary distribution for this queueing system in a symmetric case, which they obtain as a heavy traffic limit of the classical discrete model. With this as a starting point, we analyse the Brownian model directly in further detail, and consider some related exit problems.