Department of Mathematical Sciences

Dini's helix - a pseudospherical surface Brownian motion Willmore cylinder with umbilic lines
		   (Babich-Bobenko) Triadic Von Koch Snowflake - Fleckinger, Levitin
		   and Vassiliev Darboux transform of a Clifford torus (Holly
		   Bernstein) Mandelbrot fractal geometry

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Alex Cox

Prob-L@b Seminars: Spring 2010

Our seminars our usually held at 12.15 p.m. on Wednesdays in room 1W 3.6 before 19/4/2010, and 4W 1.7 after this date. If you wish to find out more, please contact one of the organisers. The speakers are mostly internal (Bath) unless otherwise stated. Details of previous semesters can be found here

17/2/10: Mathew Penrose

Covariance bounds and chaos expansion over Poisson spaces

3/3/10: Antal Járai

Zero dissipation limit in the Abelian sandpile model

10/3/10: Alexander Cox

Time-Homogeneous Diffusions with a Given Marginal at a Random Time

17/3/10: Gregorio Moreno (Paris)

Asymmetric directed polymers in random environments

We consider the point-to-point partition function of discrete time directed polymers in a random environment. We show that, when the end point is close to an axis and the temperature is properly rescaled, the free energy coincides with the one of a continuous time model in a Brownian environment. This limit is explicit in the one-dimensional case (Moriarty-O'Connell 2007). In a second time, we consider the point-to-line partition function at fixed temperature for one-dimensional directed polymers with a drift growing with the size of the box. Based on estimates for last passage percolation, we compute the free energy and give the exact order of the fluctuations.

24/3/10: Daniel Gentner (Karlsruher Institut für Technologie)

Inspecting partially stationary models of Stochastic Geometry: Theory and applications

We introduce the notion of "generalized Palm measure" which arises in partially stationary models where the underlying group operation with respect to which the model is distributionally invariant is not transitive. Though a minimal level of formality is inevitable for this introduction, our focus will then be to skip technical details and move quickly towards applications and related results. These include the recently quite popular Mass-Transport Principle, extended to non-transitive group operations, along with a quick application demonstrating its usefulness, a result about isometry-stationary tesselations on compact 2-dim Riemannian manifolds and explicit results about the distribution of "typical cells on group invariant subsets of $\mathds{R}^d$" in partially stationary or isotropic Poisson Delauney Tesselations. Here, of course, we will have to clarify the exact meaning of such "typical cells".

14/4/10: Wolfgang König (Berlin)

Phase transitions for dilute particle systems with Lennard-Jones potential

Note venue: 3E2.1

21/4/10: Gabriel Faraud (Paris XIII)

Random walk in random environment, the case of trees

The model of random walks in random environment studied by Sinaï and Solomon has received much interest over the past twenty years, and is now quite well understood. In parallel, many extensions to this model have been introduced, and some of them, in particular random walk in random environment on Zd remain quite mysterious. We will particularly focus on the case of trees. Indeed, apart from its theoretical interest, as a intermediate between dimension 1 and multidimensional models, this model presents some links with models of branching random walk, which allows to get quite precise results on its behaviour. We will try to explicit what this links are, and and give a overview of some recent results on this topic.

Note venue: 3W3.7

28/4/10: Christian Mönch

Dynamical Random Graphs with Sublinear Preferential Attachment

The Sublinear Preferential Attachment Graph is a dynamical network model whose evolution in (discrete) time is governed by a sublinear function of the degrees of its vertices. I will give an overview of results concerning main attributes of the graph like degree distribution and emergence of a giant component. If time allows, I will also prove lower bounds on the diameter for important special cases. Our model is a natural generalisation of the Barabasi-Albert model and the results support predictions of universality for a large class of random networks.

4/5/10: Tom Kurtz (Madison-Wisconsin)

Weak convergence and large deviation theory

Puhalskii, O'Brien and Vervaat, and de Acosta have developed an approach to large deviation theory that is directly analogous to the theory of weak convergence of probability measures, including a complete analog of Prohorov's theorem based on a notion of exponential tightness. This approach will be described, emphasizing the parallels with weak convergence theory. Necessary and sufficient conditions for exponential tightness for a sequence of cadlag processes in the Skorohod topology will be given that are similar to standard results for weak convergence.

Large deviations for Markov processes

For a sequence of Markov processes, convergence of Fleming's log-exponential nonlinear semigroups is shown to imply the large deviation principle in a manner analogous to the use of convergence of linear semigroups in weak convergence. In particular cases, this convergence can be verified using the theory of nonlinear contraction semigroups. The theory of viscosity solutions of nonlinear equations is used to generalize earlier results on the semigroup convergence, enabling application of the method to obtain a variety of new and known results. Control methods similar to methods presented in the book by Dupuis and Ellis are used to give representations of the rate functions.

Note time and venue: Tuesday, 10:00-12:15, 4W1.17

19/5/10: Robert Knobloch

One-sided FKPP travelling waves for fragmentation processes

We introduce the FKPP equation in the setting of fragmentation processes. Our main results are concerned with the existence and uniqueness of one-sided FKPP travelling waves. In this respect we consider a product martingale that is related to such a travelling wave solution.

26/5/10: Jiajie Wang

Root's Barriers and Partial Differential Equations: Construction and Optimality

Recent work of Dupire and Carl & Lee has emphasised the importance of understanding the Skorokhod Embedding originally proposed by Root for applications in the model-free hedging of variance options. Root's work shows that there exists a barrier from which one may define a stopping time which solves the Skorokhod embedding problem. This construction has the remarkable property, proved by Rost, that it minimises the variance of the stopping time among all solutions.
In this work, we prove a characterisation of Root's barrier in terms of the solution to a free boundary problem, originally stated by Dupire, and we give an alternative proof of the optimality property which has an important consequence for the construction of hedging strategies in the financial context.

2/6/10: Andreas Kyprianou

The prolific backbone for a super-diffusion

21/6/10: Hermann Thorisson (University of Iceland)

Mass-stationarity through the Cox process

Consider a random measure on a locally compact Abelian group, for instance the d-dimensional Euclidean space. Consider also a random element in a measurable space on which the group acts, for instance a random field indexed by the group. Mass-stationarity of the random element with respect to the measure is an intrinsic characterization of Palm versions with respect to stationary random measures. It is a formalization of the intuitive idea that the origin is a typical location in the mass of the measure. Mass-stationarity is an extension to random measures of point stationarity with respect to a simple point process.
A Cox process represents the mass of a random measure through a collection of points placed independently at typical locations in the mass. Thus if the random measure is mass-stationary and we add an extra point at the origin to the Cox process then the points of that modified Cox process are all at typical locations in the mass of the random measure. It turns out that mass-stationarity with respect to the random measure reduces to mass-stationarity with respect to the modified Cox process. In particular, for diffuse random measures mass-stationarity reduces in this way to point stationarity.

Note time and day: Monday, 14:15

15/7/10: Narn-Rueih Shieh (National Taiwan University)

The Exponential Stationary Processes

Note time and day: Thursday, 15.15


Seminars will be added to this list as they are confirmed. Please check back for the latest list, or subscribe to the prob-sem mailing list to receive details of future seminars

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