# Probabilistic approach to finite and infinite dimensional dynamical systems

PTDC/MAT/104173/2008

## Researchers

- Fernanda Cipriano
- Ana Bela Cruzeiro (PI)
- Vladimir Dragovic
- Atle Hahn
- Benjamin Himpel
- Paul Lescot
- Jean-Claude Zambrini
- Davide Masoero (post-doc)
- Miguel Tierz (post-doc)
- Xin Chen (post-doc)

## Description

One task is devoted to hydrodynamics: we do not try to introduce random noise in Euler or Navier-Stokes equations, but regard the second one as a stochastic deformation of the first equation, whose predictions should be, therefore, intrinsically probabilistic. We use the Lagrangian stochastic flows and show that the analysis of Euler equation à la Arnold can be deformed, in the same sense, when the fluid is viscous.

Another topic is centered on integrable dynamical systems and their geometry: detailed analysis of specific classical integrable systems as well as deformation of such ideas and techniques along the trajectories of diffusion processes with tools of Geometric Mechanics.

The third topic, Chern-Simons Gauge theory, includes a study from a probabilistic viewpoint. In the old tradition of Quantum Field Theory, a symbolic Feynman path integral has been the origin of an amazing number of rich geometrical consequences.

The host institution is the Group of Mathematical Physics of the University of Lisbon. This is a research centre in Mathematics funded by the Portuguese Science Foundation (FCT) which has always been awarded the highest possible classification in all international evaluations carried out by FCT; in the latest of these (2008) only 6 research units out of a total universe of 20 Maths Centres in the whole country received this classification.

The project includes funding for **two additional postdoctoral
positions** starting before the end of 2011. For a description
of the conditions, deadlines, etc, please see the call for applications.

Time span: 01/01/2010-31/12/2012

Funding institution: FCT

Budget: € 122261.00

## Some relevant publications

(for other publications by the researchers involved, see the respective homepages)

- F. Cipriano and A. B. Cruzeiro
- Navier-Stokes equation and diffusions on the group of
diffeomorphisms of the torus

*Commun. Math. Phys.***275**(2007), 255-269 - V. Dragovic and M. Radnovic
- Hyperelliptic Jacobians as billiard algebra of pencils of quadrics:
beyond Poncelet porisms

*Advances in Math.***219**(2008), 1577-1607 - P. Lescot and J.-C. Zambrini
- Probabilistic deformation of contact geometry, diffusion processes
and their quadratures

*Progress in Probab. Birkhäuser***59**(2007), 203-226 - A. Hahn
- Chern-Simons models on S²×S¹, torus gauge fixing and link
invariants II

*J. Geom. Phys.***58**(2008), 1124-1136