## Tuesdays 4.30-5.30 pm,
MR 12.
Webpage of the probability group Tea beforehand at 4pm in the common room, and drinks afterwards at the Punter. |

Date | Speaker | Title | Notes |

18 January | Laurent Saloff-Coste (Cornell) | Time inhomogeneous Markov chains: merging and stability (abstract) | |

25 January | Johannes Ruf (Columbia) | Hedging under arbitrage (abstract) | Special time: 2pm. |

25 January | Antal Jarai (Bath) | Abelian sandpiles on infinite graphs (abstract) | |

26 January | Mark Kelbert (Swansea) | Continuity of mutual information and the entropy-power inequality (abstract) | Additional seminar, unusual day and time. |

1 February | Daniel Ueltschi (Warwick) | Probabilistic representation of quantum correlations (abstract) | |

8 February | Nicolas Fournier (Marne-La-Vallée) | Scaling limits of forest fire processes. (abstract) | |

15 February | Remi Peyre (Lyon) | A probabilistic approach to Carne's bound (abstract). | |

22 February | Wendelin Werner (Univerité Paris-Sud and ENS) | Self-interacting random walks and their asymptotic behavior (abstract). | |

1 March | Yuri Suhov (Cambridge). | New results on spectra of multi-particle Schrödinger operators with random potentials (abstract). | |

8 March | Julien Berestycki (Paris VI) | The branching Brownian motion seen from its tip (abstract). | |

15 March | Hugo Dominil-Copin (Geneva) | Critical temperature of the square lattice Potts model (abstract). |

We consider the problem of obtaining quantitative results regarding the behavior of well-behaved time inhomogeneous Markov chains on finite state spaces. The problem is difficult (Joint work with J. Zuniga).

Explicit formulas for optimal trading strategies in terms of minimal required initial capital are derived to replicate a given terminal wealth in a continuous-time Markovian context. To achieve this goal this talk does not assume the existence of an equivalent local martingale measure. Instead a new measure is constructed under which the dynamics of the stock price processes simplify. It is shown that delta hedging does not depend on the ``no free lunch with vanishing risk'' assumption. However, in the case of arbitrage the problem of finding an optimal strategy is directly linked to the non-uniqueness of the partial differential equation corresponding to the Black-Scholes equation. The recently often discussed phenomenon of ``bubbles'' is a special case of the setting in this talk.

The Abelian sandpile model was introduced in the physics literature as a toy model of "self-organized criticality". Originally, it is defined as a Markov chain taking place on particle configurations on a finite graph. It received a lot of attention, since several of its characteristics, for example spatial correlations, were observed to follow power laws, akin to critical systems in statistical physics. In this talk, I will give an overview of results about limits on certain infinite graphs, including the d-dimensional integer lattice. I will also consider a perturbation of the model that has exponential decay of correlations, with rate of convergence estimates as the perturbation parameter vanishes. (In large part this is based on joint works with S.R. Athreya, R. Lyons, F. Redig and E. Saada.

Consider an additive channel of information transmission with a continuous noise. The talk will focus on sufficient conditions for continuity of the input-output mutual information for large and small signal-to-noise ratios. This result leads to Shannon's entropy-power inequality which is a far-reaching generalisation of Minkowski's inequality from the classical analysis.

I this talk I will speak about a 1985 result from Carne & Varopoulos: consider a Markov chain on a graph (i.e. whose transitions always follow the edges) which is reversible with stationary measure $\mu$; then, denoting by $p^t(x,y)$ the probability that a chain starting at $x$ is at $y$ at time $t$, one has the Gaussian bound \[ p^t(x,y) \leq \sqrt{e} \big( \mu(y)/\mu(x) \big)^{1/2} \exp \big( -d(x,y)^2 / 2t \big) ,\] where $d(x,y)$ is the graph distance between $x$ and $y$. My goal will be to explain this result by probabilistic arguments (which was not the case of the original proof), close to the forward/ backward martingale decomposition. This approach will lead to a generalization of Carne's bound to the case where the particle can occasionally make a jump not following an edge. I will also explain how one can improve Carne's bound by a spectral factor, by considering the chain "conditioned to be recurrent".

Random Schrödinger operators are popular in solid-state quantum physics because they provide accurate mathematical models for transitional phenomena occurring at the border between ordered and disordered systems (localisation {\it vs} delocalisation of eigenstates). Mathematically, the problem is to analyse properties of the spectrum of such an operator: this presents a challenge since new ideas and methods are required, combining Probability, Functional Analysis and -- occasionally -- other mathematical disciplines.

In this talk, I'll discuss new results on multi-particle random Schr\"odinger operators in a Euclidean space (joint works with several colleagues completed in 2010). The main result is that when the randomness is `strong', the spectrum near its lower edge is pure point with probability one and the corresponding eigenfunctions are exponentially localised.

No preliminary knowledge of Quantum Mechanics will be assumed.