# East Midlands Stochastic Analysis Seminar

**Thursday 29th - Friday 30th August 2013**

**PROGRAMME**

**Thursday 29th August Loughborough University, Room W002, Sir David Davis Building
**12:30-14:00 Lunch (at Elvyn Richards)

14:00-15:00

**Ian Melbourne**(Warwick)

*Interpretation of stochastic integrals that arise as limits of deterministic systems via homogenization*

15:00-15:30 Tea

15:30-16:30

**Wenlian Lu**(Fudan University and CS Warwick)

*Synchronisation in networks of coupled systems with stochastically time-varying couplings*

16:30-17:30

**Simon Harris**(Bath)

*Branching Brownian motion with killing*

19:00 Evening meal

**Friday 30th August Warwick University, Mathematics Institute, Room MS.B3.02
**10:00-11:15 Coffee in the Mathematics Institute Common Room

11:30-12:30

**Robin Hudson**(Loughborough)

*Quantum probability from Heisenberg uncertainty to Lévy area*

12:30-14:00 Lunch at Le Gusta, Arts Centre Building

14:00-15:00

**Michel Emery**(Strasbourg)

*On parametrizations of filtrations in negative, discrete time*

15:00-15:30 Tea in the Mathematics Institute Common Room

15:30-16:30

**Chunrong Feng**(Loughborough)

*Random Periodic Solutions*

16:30-17:30

**Hiroyuki Matsumoto**(Aoyama Gakuin University)

*First hitting times of Bessel processes and zeros of modified Bessel functions*

17:30 Drinks in the Mathematics Institute Common Room

19:00 Evening meal

ABSTRACTS

Wenlian Lu (Fudan University and CS Warwick)

Title: *Synchronisation in networks of coupled systems with stochastically time-varying couplings*

Abstract: Synchronisation problem has been recognised as being of importance in distributed coordination of dynamic agent systems, which is widely applied in distributed computing, management science, flocking/swarming theory, distributed control, and sensor networks. In the past decades, stability analysis of synchronisation in networks of coupled dynamical systems has been one of foci in control theory and applied mathematics. Besides static network topology, in many real-world applications, the agents are moving with randomness. In this case, stochastically time-varying topology under link failure or creation should be considered. In this talk, I would like to present the recent works on stability analysis of synchronisation in coupled systems with stochastically time-varying diffusion terms by a combination of dynamical system theory and stochastic analysis. The main mathematic techniques include the Hajnal diameter that stands for the distance between dynamics of nodes and was extended for stochastic cases and algebraic graph theory that describes the information community between nodes. These ideas can be generalised as transverse stability of random dynamical systems.

Ian Melbourne (Warwick)

Title:* Interpretation of stochastic integrals that arise as limits of deterministic systems via homogenization*

Abstract: Homogenization is a mechanism whereby multiscale deterministic systems converge to stochastic differential equations. In achieving a rigorous theory, the key question is the interpretation (Stratonovich, Ito, other) of the stochastic integrals present in the limit. In this talk, we present a definitive solution to this question, thereby shedding light on the general question of how to correctly interpret stochastic integrals arising via smooth approximation.

This is joint work with David Kelly.

Robin Hudson (Loughborough)

Title: *Quantum probability from Heisenberg uncertainty to Lévy area*.

Abstract: Quantum probability is a generalization of the classical Kolmogorovian theory, in which random variables have a noncommutative multiplicative structure. It is distinguished from free probability theory by being closely motivated by the commutation relations of quantum physics, such as the Heisenberg commutation relation pq-qp=-ih/2/ /pi for momentum and position. Notions such as independence are less exotic than in free probability, so that it is natural to construct Brownian motions each of which is individually fully equivalent to the Wiener process, but which do not commute with each other, which jointly obey a noncommutative extension of Ito^^ calculus, and which have a mutual independence property allowing them to be used to construct a quantum Le^'vy area.

In this talk I shall develop quantum probability from its origins in mathematical physics, through a quantum central limit theorem whose functional form leads to such noncommuting Brownian motions, and I will describe a counting problem arising from the quantum Lévy area formula whose solution is found in terms of Catalan numbers and derivatives from them.