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

Informal Probability Seminars: Winter 2008

Our seminars our usually held at 12.15 p.m. on Wednesdays in room 1W 3.6 . 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

8/10/08: Ron Doney (Manchester)

Convergence of local times of Levy processes

The following problem is discussed: if a sequence of Levy processes (or normed random walks) converges to a limiting Levy process, does the corresponding sequence of local times of the reflected processes converge?

15/10/08: Peter Mörters

Two remarks on a paper by Kallenberg

In a recent discussion paper, Kallenberg observed a great similarity in the local hitting, scaling, and conditioning behaviour of several random measures, including simple point processes, local times of regenerative sets and the states of superprocesses. In this talk I would like to add two further observations: First, the intersection local times of Brownian paths also share the common features of these random measures, and second, all these measures also have strikingly similar multifractal behaviour. I will also discuss similarities in the Hausdorff and packing measure properties of their supports.

29/10/08: Vadim Shcherbakov

Stability of a growth process generated by cooperative sequential adsorption

We are interested in a long time behaviour (stability) of a growth process generated by the cooperative sequential adsorption (CSA)model .
The CSA model is a probabilistic model of adsorption processes. In the talk, we consider the CSA model on a lattice, which in turn can be viewed as a variant of the Polya urn scheme with interactions: namely, we sequentially attach particles to lattice sites with probabilities which depend on the configuration of already allocated particles.
The growth process is then described by the numbers of attached (adsorbed) particles. Stability of the growth process now means, loosely speaking, that its components grow at approximately the same rate, therefore, for instance, no extraordinary high peaks are observed. For different models of attachment configurations, we rigorously prove the stability (or instability) of the process.
This is a joint work with Stanislav Volkov (University of Bristol).

5/11/08: Andreas Kyprianou

Refracted Lévy processes

12/11/08: Nikolai Leonenko (Cardiff)

Multifractality of products of geometric Ornstein-Uhlenbeck type processes

This is joint work with V.V. Anh (Queensland University of Technology, Brisbane) and N.-R. Shieh (National Taiwan University, Taipei). We consider multifractal products of stochastic processes as defined in Mannersalo et al. (2002), but we provide a new interpretation of the conditions on the mean, variance and covariance functions of the resulting cumulative processes in terms of the moment generating functions. We show that the logarithms of the corresponding limiting processes have an infinitely divisible distribution such as the gamma and variance gamma distributions (resulting in the log-gamma and log-variance gamma scenarios respectively), the inverse Gaussian and normal inverse Gaussian distributions (yielding the log-inverse Gaussian and log-normal inverse Gaussian scenarios respectively). We describe the behavior of their q-th order moments and Rényi functions, which are non-linear, hence displaying their multifractality. A property on the dependence structure of the limiting processes, leading to their possible long-range dependence, is also obtained. We consider very different scenarios such as the log-gamma and log-inverse Gaussian scenarios as typical examples of our approach. We should also note some related results by Barndorff-Nielsen and Schmiegel (2004) who introduced some Lévy-based spatiotemporial models for parametric modelling of turbulence. Log-infinitely divisible scenarios related to independently scattered random measures were introduced in Bacry and Muzy (2003) and others. We should note that Chris Heyde (1999) proposed to use a multifractality into risky asset model with fractal activity time (see also Heyde and Leonenko (2005)).

19/11/08: Hubert Lacoin (Paris)

Directed random polymer on a diamond lattice.

(Joint work with Gregorio Moreno, supervised by Pr. Francis Comets and Pr. Giambattista Giacomin)
A great number of model in statistical physics (e.g. Ising model, Potts Model, first/last passage percolation, wetting/pinning models) have been studied on a certain class of hierarchical graphs: the diamond lattices. The main advantage of these lattices is that the renormalisation methods that are usually used by physicists can be performed in a rigourous manner. In this talk, I will present some results obtained with Gregorio Moreno for the random polymer model on the diamond lattice and the different tools that are used for the proofs (fractional moment method, shift of the environment, percolation argument and second moment method) and discuss about the similarities of this model with the one on $Z^{d+1}$.

3/12/08: Juan Carlos Pardo Millan

Exact and asymptotic n-tuple laws at first and last passage

Understanding the space-time features of how a Lévy process crosses a constant barrier for the first time, and indeed the last time, is a problem which is central to many models in applied probability such as queueing theory, financial and actuarial mathematics, optimal stopping problems, the theory of branching processes to name but a few. In Kyprianou and Doney a new quintuple law was established for a general Lévy process at first passage above a fixed level. In this work, we use the quintuple law to establish a family of related joint laws, which we call n-tuple laws, for Lévy processes, Lévy processes conditioned to stay positive and positive self-similar Markov processes at both first and last passage over a fixed level. Here the integer n typically ranges from three to seven. Moreover, we look at asymptotic overshoot and undershoot distributions and relate them to overshoot and undershoot distributions of positive self-similar Markov processes issued from the origin. Although the relation between the n-tuple laws for Lévy processes and positive self-similar Markov processes are straightforward thanks to the Lamperti transformation, by inter-playing the role of a (conditioned) stable processes as both a (conditioned) Lévy processes and a positive self-similar Markov processes, we obtain a suite of completely explicit first and last passage identities for so-called Lamperti-stable Lévy processes. This leads further to the introduction of a more general family of Lévy processes which we call hypergeometric Lévy processes, for which similar explicit identities may be considered.
(Joint work with A.E. Kyprianou and V. Rivero)

10/12/08: Jochen Voss (Warwick)

An SPDE-based Sampling Method

In many situations it is useful to be able to simulate paths from an SDE conditioned on some interesting event occurring. But, while simulating an unconditioned SDE is often trivial, simulating paths from a conditioned SDE can be a challenge. In this talk we present a new sampling method to generate paths from the distribution of a second order SDE under end-point constraints.
The proposed method works by constructing a fourth-order SPDE which is ergodic and has the distribution of the conditioned SDE as its invariant measure ("space" of the SPDE corresponds to "time" of the SDE). Paths of the conditioned SDE can then be found by simulating the sampling SPDE until it is close to stationarity and statistical properties of the conditioned SDE can be found by taking ergodic averages of the solution of the SPDE. In contrast to earlier works (were we used a second order SPDE to sample for the distribution of a conditioned first order SDE) the present method also allows to consider the case where the drift in the original SDE does not have a gradient structure.
We illustrate the method with the help of numerical simulations of the resulting SPDE, obtained by using finite elements discretisation.
(Joint work with Martin Hairer and Andrew Stuart.)


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