Oxford-Man Institute

We present a new class of self interacting Markov chain models. In contrast to traditional Markov chains, their time evolution may depend on the occupation measure of the past values. We propose a theoretical basis based on measure valued processes and semigroup technics to analyze their asymptotic behaviour as the time parameter tends to infinity. We exhibit different types of decays to equilibrium depending on the level of interaction. In the end of the talk, we shall present a self interacting methodology to sample from a sequence of target probability measures of increasing complexity. We also analyze their fluctuations around the limiting target measures.

We consider transient random walks in random environment on Z with zero asymptotic speed. In a seminal paper, Kesten, Kozlov and Spitzer proved that the hitting time of the level "n" converges in law, after a proper normalization, towards a positive stable law, but the question of the description of its parameter was left open since that time. A new approach to this problem, based on a precise description of Sinai's potential, leads to a complete characterization of this stable law, making a tight link with Kesten's renewal series. The case of Dirichlet environment turns out to be remarkably explicit. Quenched results on this model will be presented if time permits.

We present numerical schemes for nonlinear stochastic differential equations whose solution evolves on a smooth finite dimensional manifold. Given a Lie group action that generates transport along the manifold, we pull back the stochastic flow on the manifold to the Lie group via the action and subsequently to the corresponding Lie algebra.

We construct an approximation to the stochastic flow in the Lie algebra via closed operations and then push back to the manifold, thus ensuring our approximation lies in the manifold. We call such schemes stochastic Munthe-Kaas methods after their deterministic counterparts. We also present stochastic Lie group integration schemes based on Castell--Gaines methods. They become stochastic Lie group integrator schemes if we use Munthe-Kaas methods as the underlying ordinary differential integrator. Lastly, we demonstrate our methods by presenting some numerical examples

We will examine the typical structure of random polytopes by projecting the three fundamental regular polytopes: the simplex, cross-polytope, and hypercube. Along the way we will explore the implications of their structure for information acquisition and optimization. Examples of these implications include: that an N-vector with k non-zeros can be recovered computationally efficiently from only n random projections with n=2e k log(N/n), or that for a surprisingly large set of optimization problems the feasible set is actually a point. These implications are driving a new signal processing paradigm, Compressed Sensing, which has already lead to substantive improvements in various imaging modalities. This work is joint with David L. Donoho.

We will consider a (sub) critical Galton-Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We shall specify the law of this allelic partition in terms of the distribution of the number of clone-children and the number of mutant-children of a typical individual. Some limit theorems related to the distribution of the allelic partition will be also presented.

We modify the usual Erdos-Renyi random graph evolution by letting connected clusters 'burn down' (i.e. fall apart to disconnected single sites) due to a Poisson flow of lightnings. In a range of the intensity of rate of lightnings, the system sticks to a permanent critical state (i.e. exhibits so-called self-organised critical behaviour). The talk will be based on joint work with Balint Toth.

The uniform spanning forest (USF) in a graph

is a random spanning forest obtained as the limit of uniformly chosen spanning

trees on finite subgraphs. The USF is known to have stochastic dimension 4 on

graphs that are "at least 4 dimensional" in a certain sense. In this

talk I will look at more detailed estimates on the geometry of a fixed

component of the USF in the special case of the d-dimensional integer lattice,

d > 4. This is motivated in part by the study of random walk restricted to a

fixed component of the USF.

The system

u_t = Delta u + buv - cu + u^{1/2} dW

v_t = - uv

models the evolution of a branching population and its usage of a non-renewable resource.

A phase diagram in the parameters (b,c) describes its long time evolution.

We describe this, including some results on asymptotics in the phase diagram for small and large values of the parameters.

We introduce a class of backward stochastic differential equations (BSDEs) driven by Brownian motion and Poisson random measure, and subject to constraints on the jump component. We prove the existence and uniqueness of the minimal solution for the BSDEs by using a penalization approach. Moreover, we show that under mild conditions the minimal solutions to these constrained BSDEs can be characterized as the unique viscosity solution of quasi-variational inequalities (QVIs), which leads to a probabilistic representation for solutions to QVIs. Such a representation in particular gives a new stochastic formula for value functions of a class of impulse control problems. As a direct consequence we obtain a numerical scheme for the solution of such QVIs via the simulation of the penalized BSDEs. This talk is based on joint work with I. Kharroubi, J. Ma and J. Zhang.