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 2011

Our seminars our usually held at 12.15 p.m. on Wednesdays in room 4W 1.7. 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

9/2/11: Ben Hambly (Oxford)

Spectral properties and random walk on the critical random graph

16/2/11: Markus Riedle (Manchester)

Cylindrical Lévy processes in Banach spaces

The objective of this talk is the introduction of cylindrical Lévy processes and their stochastic integral in Banach spaces.
The degree of freedom of models in infinite dimensions is often reflected by the request that each mode along a dimension is independently perturbed by the noise. In the Gaussian setting, this leads to the cylindrical Wiener process including from a model point of view the very important possibility to model a Gaussian noise in both time and space in a great flexibility (space-time white noise). Up to very recently, there has been no analogue for Lévy processes.
Based on the theory of cylindrical processes and cylindrical measures we introduce cylindrical Lévy processes as a natural generalisation of cylindrical Wiener processes. We continue to characterise the distribution of cylindrical Lévy processes by a cylindrical version of the Lévy-Khintchine formula.
The cylindrical approach enables the definition of a stochastic integral with respect to a cylindrical Lévy process without any geometric constraints on the underlying Banach space. We use this integral to develop a theory of cylindrical stochastic Cauchy problems and demonstrate its practicalness by presenting some basic facts on the cylindrical Ornstein-Uhlenbeck process driven by a cylindrical Lévy process.
(part of this talk is based on joint work with D. Applebaum)

23/2/11: Peter Mörters

Distances in ultrasmall random networks

Scale free networks are ubiquitous in our lives, appearing in the form of social or communication networks, collaboration networks, or the world-wide web. One of the central observations in the theory of scale-free random networks is that in the case of power-law exponents τ ∈ (2,3) networks are ultrasmall, which means that the distance of two randomly chosen nodes in the giant component of a graph with n vertices is of asymptotic order log log n. We refine this observation and argue that these networks can be further divided into two universality classes corresponding to quite different internal architecture. The talk is based on joint work with Steffen Dereich (Marburg) and Christian Moench (Bath).

1/3/11: Ron Doney (Manchester)

First passage times for random walks and Lévy processes

The behaviour of the tail of the distribution of the first passage time over a fixed level has been known for many years, but until recently little was known about the behaviour of the probability mass function or density function. In this talk we describe recent results of Vatutin and Wachtel, Doney, and Doney and Rivero which give such information whenever the random walk or Lévy process is asymptotically stable.

Note day and venue: Tuesday, 3W 3.7

9/3/11: Daisuke Shiraishi (Kyoto)

Random walks on the random walk trace

Let S be a simple random walk starting at the origin in Z4. We consider G = S[0,∞) to be a random subgraph of the integer lattice and assume that a resistance of unit 1 is put on each edge of the graph G. Let Rn be the effective resistance between the origin and Sn. We derive the exact value of the resistance exponent; more precisely, we prove that n-1 E(Rn) (log n)-1/2. Furthermore, we derive the precise exponent for the heat kernel of a random walk on G at the quenched level. These results give the answer to the problem raised by Burdzy and Lawler (1990) in four dimensions.

23/3/11: Martin Kolb (Oxford)

Spectral Analysis of Diffusions with Jump Boundary

In the talk we describe recent results concerning spectral properties of so called diffusions with jump boundary. Though these processes and their generators already have been constructed by Feller, the investigation of their large time behaviour is much more recent. In the talk results of Pinsky/Ben-Ari, Leung/Li/Rakesh and Kolb/Wuebker will be presented

29/3/11: Pierre Patie (Université Libre de Bruxelles)

Exponential Functional of Lévy Processes

The law of the exponential functional of Lévy processes plays a prominent role from both theoretical and applied perspectives. We start this talk by describing some reasons motivating its study and we review all known results concerning the distribution of this random variable. We proceed by describing a new factorization identity for the law of the exponential functional under very mild conditions on the underlying Lévy process. As by-product, we provide some interesting distributional properties enjoyed by this random variable as well as some new analytical expressions for its distribution (Joint work with J.C. Pardo (CIMAT, Mexico) and M. Savov (The University of Oxford, UK)).

Note time, day and venue: Tuesday, 14.15, 6E 2.2

30/3/11: Curdin Ott

Russian Options with a Cap

We are interested in a modification of the optimal stopping problem associated with the pricing of Russian options in financial markets whose underlying process is an exponential spectrally negative Lévy process (Ann. App. Prob., 2004, Vol. 14, No.1, pp. 215-238). It turns out that the optimal stopping boundary can be characterised by a simple ordinary differential equation involving the scale functions associated with the given Lévy process. In particular, it is non-flat and varies according to the path variation of the Lévy process. As an application, we determine (in terms of scale functions) the fair price of Russian options with a cap, that is, Russian options whose payoff is bounded from above by some constant.

6/4/11: Yogeshwaran Dhandapani (École Normale Supérieure)

Percolation and directionally convex ordering of point processes.

In this talk, we explain the relation between directionally convex ordering of point processes and percolation. Directionally convex ordering has been used to compare point processes with same mean intensities. We will start with a primer on directionally convex ordering of point processes and examples of point processes that are directionally convex ordered. We link directionally convex ordering to percolation as well as clustering by showing that they impact negatively the capacity functionals of their corresponding Boolean models. This is used to show ordering of some new critical radii which act as upper and lower bounds to the usual critical radius for percolation of a point process. The upper bound increases with dcx order while the lower bound decreases.

In the second part, we exploit the fact that many probabilities of additive shot-noise fields of point processes can be bounded by their Laplace transforms. This for sparse point processes (i.e, lesser than Poisson point process in dcx order) can be bounded by the corresponding Laplace transform of Poisson-driven shot-noise fields. For a nice class of functionals, one can compute the latter explicitly to ascertain non-triviality of phase transition in various percolation models. We carry out such a program for providing uniform upper and lower bounds (uniform over all sparse point processes) for the critical radius for k-percolation and percolation in SINR (Signal-to-Interference-Noise-Ratio) graphs.

Note venue: 1W 2.5

4/5/11: Albert Ferreiro-Castilla (Barcelona)

Inversion of analytic characteristic functions and infinite convolutions of exponentials

11/5/11: Xiong Jin(St Andrews)

Dimension result for two-dimensional multiplicative cascade processes

I will present a Hausdorff dimension result for the image of two-dimensional multiplicative cascade processes, and obtain from this result a KPZ-type formula which normally has one point of phase transition.

Note venue: 3W 4.7

12/5/11: Günter Last (Karlsruhe)

Minicourse I: Point processes and random measures

Minicourse: Topics in Stochastic Geometry
Stochastic geometry aims to develop and to analyze mathematical models for random spatial patterns. This mini-course provides an introduction into some topics of stochastic geometry. The lectures will not only cover some classical results for stationary tessellations and the (Poisson) Boolean model but will also present some new developments on invariant transports of random measures and continuum percolation.

Note time and venue: 16.15, 3W 3.7

13/5/11: Günter Last (Karlsruhe)

Minicourse II: Random Tesselations

Note time and venue: 16.15, 3W 3.7

18/5/11: Günter Last (Karlsruhe)

Minicourse III: Random partitions and balanced invariant transports

Minicourse IV: The Boolean model

Note time: 10.15-12.15

18/5/11: Thomas Dusquene (Paris VI)

General Growth processes of trees

We introduce the notion of a hereditary property for discrete trees and study associated forest growth processes hence providing a unified approach to various reduction and growth procedures of Galton-Watson trees that have been studied previously. We shall prove that the only possible limits are Lévy trees. This combinatorial approach provide an easy characterisation of Lévy trees.
This is a joint work with Matthias Winkel.

19/5/11: Günter Last (Karlsruhe)

Minicourse V: Percolation on planar tessellations

Note time: 16.15

25/5/11: Neil O'Connell (Warwick)

Generalizations of Pitman's 2M-X theorem and their applications

Pitman's 2M-X theorem states that if Xt is a standard one-dimensional Brownian motion and Mt = maxs ≤ t Xs then 2M-X is a three-dimensional Bessel process. This theorem has vast generalizations. I will describe some of these including a particular generalization involving exponential functionals of Brownian motion which was discovered by Matsumoto and Yor, and a multi-dimensional version of this which is related to the quantum Toda lattice (this will be explained) and has applications to a Brownian directed polymer model.

8/6/11: Sergey Bocharov

Branching Brownian Motion with branching at the origin

We shall discuss a Branching Brownian Motion model, where branching takes place at the origin. In this model particles split at rate β on the local time scale. We shall present results about the almost sure asymptotic population growth as well as the spatial spread of the system.

14/6/11: Elie Aidekon (TU Eindhoven)

The extremal process of the branching Brownian motion

We look at the branching Brownian motion on the real line. Particles move independently according to a Brownian motion and split into two at exponential times. At time t, we are interested in the 'landscape' seen from the particle located at the minimum of this process. What does the point process consisting of the particles around the minimum look like? We prove a convergence in law of this process, and give a description in terms of a decorated Poisson point process. Joint work with J. Berestycki, E. Brunet and Z. Shi.

Note day: Tuesday

15/6/11: Samuel Cohen (Oxford)

Nonlinear expectations and BSDEs in general probability spaces

Much work has been done on time-consistent risk-averse decision making. One approach is to axiomatically define a filtration consistent nonlinear expectation, which is a family of operators satisfying many of the properties of the conditional expectation, but can be nonlinear. This axiomatic approach raises the question of how to construct these expectations -- a key approach is to define them using solutions to Backward Stochastic Differential Equations (BSDEs), but which nonlinear expectations can be defined in this way?

We consider nonlinear expectations in probability spaces satisfying only the usual conditions and separability. We give a form of BSDE in these spaces, and show existence and uniqueness of solutions, and also a comparison theorem. We demonstrate that all nonlinear expectations satisfying a domination assumption can be expressed as the solutions to BSDEs with Lipschitz continuous drivers where the comparison theorem holds, extending the results of Coquet, Hu, Memin and Peng (2002) to general probability spaces.

6/7/11: Piotr Milos (Warsaw)

CLT for an Ornstein-Uhlenbeck branching system


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