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## Vidi project Stationary dynamics in infinite dimensions |

## general |

The project is supported by the Netherlands Organisation for Scientific Research (NWO). It started 1 January 2006 and will end on 31 December 2010.

It is a fact of life that many dynamic phenomena in physics, engineering, economics, finance, biology, life sciences, and other areas can adequately be modelled by differential or difference equations. In many instances there is an influence of noise that cannot be neglected. It is often an important question how such an evolutionary system will behave in the long run. Does the behavior become stationary when the exterior forces have settled? How will the stationary behavior then be? The project focusses on two types of stationary behavior.

The first one is existence of stationary distributions for stochastic evolution equations. The powerful functional analytic method for deterministic equations will be combined with the latest developments in stochastic integration to develop a theory for evolution equations perturbed by stochastic noise. The research will build upon the functional analytic approach for semilinear stochastic parabolic equations. The project aims to provide powerful new methods to analyze also other classes of equations (such as equations with delayed feedback), more general noise processes, and more general nonlinearities. Thus the project will provide tools to analyze, for instance, the dynamics of structured population models.

The second type of stationary behavior that will be studied is periodicity. Questions about periodicity arize, for instance, in discrete event systems and Boolean networks, which are, for example, used to analyze biochemical processes. With a suitable choice of state space they can be formulated as nonexpansive dynamical systems. There is a close relationship between the periods of a nonexpansive dynamical system and the geometric structure of its state space. It is the aim to clarify this relationship and in particular to describe the spaces in which only relatively small periods are possible. The research will be based on isometric structures rather than monotonicity assumptions.

## team |

Joris Bierkens, PhD student

Abdelhadi Es-Sarhir, postdoc

Igor Stojkovic, PhD student

Onno van Gaans, project leader

## participation in "Jump-diffusion theory for scenario evaluation of bond portfolios" |

Since October 2007, I collaborate with Dirard Mikdad M.Sc. from Rabobank in his project "Jump-diffusion theory for scenario evaluation of bond portfolios". We wish to apply state-of-the-art theory of (infinite dimensional) jump-diffusion models to reliable and efficient scenario testing of bond portfolios.

## past activities |

- 2008/12/13-20: visit to Anke Kalauch (TU Dresden)

- 2008/11/14: lecture by Igor Stojkovic in the AiO Seminar Mathematics at the University of Amsterdam, title "Gradient flows on metric spaces, and the Wasserstein space of probability measures", 16:00-17:00h.

- 2008/10/31: last working day of Abdelhadi Es-Sarhir on the VIDI project. He continues his research as Postdoctoral researcher at the Technische Universitaet Berlin.

- 2008/09/28-10/04: visit of Anke Kalauch (TU Dresden).

- 2008/09/19: Ton van Boxtel has defended his Master thesis A Banach Space-Valued Stochastic Integral with respect to a Jump Process.

- 2008/07/22-29: visit to Markus Riedle (University of Manchester).

- 2008/06/22-28: Igor Stojkovic attended the C.I.M.E. Summer Course Nonlinear partial differential equations and applications in Cetraro (Cosenza), Italy.

- 2008/06/5-8: Abdelhadi Es-Sarhir visited Prof. Zegarlinski at Imperial College, London, and gave a lecture on Friday 6 June.

- 2008/06/3-7: visit to Bas Lemmens (University of Warwick).

- 2008/05/22-29: visit of Markus Riedle (University of Manchester).

- 2008/04/07: lecture in the Analysis Colloquium of the TU Delft, title "Stochastic integration in Banach spaces relative to Levy processes", 16:00-17:00h.

- 2008/04/03: lecture in Analysis seminar at Leiden University, title "Periods of nonexpansive maps on finite dimensional normed spaces", 13:00-14:00h.

- 2008/03/31-04/4: Igor Stojkovic participated in the Workshop Aspects of Optimal Transport in Geometry and Calculus of Variations at the University of California in Los Angeles, CA, USA.

- 2008/02/27: lecture in the General mathematics colloquium at the Radboud University in Nijmegen, title "Periods of nonexpansive maps on finite dimensional normed spaces", 11:00-12:00h.

- 2008/02/4-8: Joris Bierkens participated in the workshop Harmonic Analysis and Evolution Equations, GNAMPA, in Parma, Italy.

- 2008/01/23-27: visit to Anke Kalauch (TU Dresden).

- 2007/10/3-6: Abdelhadi Es-Sarhir participated in the workshop Evolution equations and Kolmogorov operators at University of Salerno at Salerno, Italy.

- 2007/09/12-19: visit of Anke Kalauch (TU Dresden).

- 2007/08/6-10: Joris Bierkens, Igor Stojkovic and Onno van Gaans participated in the conference Stochastic Processes and their Applications 2007 at University of Illinois at Urbana-Champaign. Lecture "Stationary solutions of stochastic delay differential equations driven by Levy noise" by Onno van Gaans on Fri 10 August, 14:00-14:25h. Lecture "Semimartingale properties of stochastic convolutions" by Igor Stojkovic on Fri 10 August, 14:30-14:55h.

- 2007/07/23-27: participation in the conference Positivity V at Queen's University Belfast, lecture "On non-expansive maps dynamics and disjointness reflecting norms", Thu 26 July, 9:50-10:20h.

- 2007/05/15-22: visit to Anke Kalauch (TU Dresden).

- 2007/03/15: lecture by Igor Stojkovic in the Analysis seminar of Leiden University, title "Semimartingale properties of stochastic convolutions", 13:00-14:00h.

- 2007/03/14-20: visit of Anke Kalauch (TU Dresden).

- 2007/02/19-21: visit of Gerard Buskes (University of Mississippi).

- 2007/01/29-02/02: work visit to TU Darmstadt, Angewandte Analysis (Arbeitsgruppe 4), lecture in the Darmstaedter Analysis-Tag, title "Invariant measures for stochastic delay equations" on Tue 30 January, 10:00-10:35h.

- 2006/11/28-12/3: visit to Anke Kalauch (TU Dresden).

- 2006/11/22-26: work visit to Markus Riedle (Universitaet Mannheim), lecture in the seminar Stochastische Differentialgleichungen, title "Stationaere Loesungen von stochastischen Differentialgleichungen mit Delay" on Thu 23 November, 12:00-13:00h.

- 2006/11/16: lecture by Joris Bierkens in the Analysis seminar of Leiden University, title "The generator of stochastic delay differential equations", 13:00-14:00h.

- 2006/11/1: lecture in General Mathematics Colloquium of the University of Amsterdam, title "Invariant measures for infinite dimensional stochastic differential equations", 11:15-12:15h.

- 2006/8/6-12: work visit to Anke Kalauch (TU Dresden)

- 2006/6/8-11: participation in the workshop and conference "Carthapos 2006" in Carthage, Tunisia, lecture "Extension and restriction of ideals and bands in pre-Riesz spaces" on Fri 6 June, 11:10-11:30h

- 2006/5/31: last session of the seminar "Evolution Equations in Probability Spaces and the Continuity Equation"

- 2006/5/22-27: participation in the conference "Banach spaces and their applications in analysis" at Miami University, Oxford, Ohio, lecture "One-complemented subspaces of finite dimensional vector spaces with smooth lattice norms" on Fri 26 May, 16:40-17:10h

- 2006/3/5-11: visit of Abdelhadi Es-Sarhir (TU Darmstadt), lecture "A functional analytic approach to transition semigroups in infinite dimensions" in the Analysis and stochastics seminar, Thu 9 March, 13:00-14:00h

- 2006/3/1: lecture "Invariant measures for infinite dimensional stochastic differential equations" in the Stafcolloquium of the Mathematical Institute (Leiden), 16:00-17:00.

- 2006/2/18-26: visit of Anke Kalauch (TU Dresden), lecture "M-operators on ordered normed spaces"

- 2006/2/15: lunch lecture for De Leidsche Flesch (the student association for mathematics, computer science, and the natural sciences of Leiden University), "Stochastische differentiaalvergelijkingen in oneindigdimensionale ruimten", 13:15 - 13:45.

- 2006/2/7: presentation in the This week's discoveries seminar (Leiden University), "Stationary dynamics in infinite dimensions", 13:20 - 13:40. Here is a video of the presentation.

- 2006/2/1: first session of the seminar "Evolution Equations in Probability Spaces and the Continuity Equation", continues weekly on Wednesday afternoon from 14:15 to 17:00

- 2006/1/1: start of my Vidi-project

- 2005/10/11: An article about my Vidi-project appeared in the Nieuwsbrief van 11 oktober 2005 (in Dutch, the news letter of Leiden University).

## production |

- Markus Riedle and Onno van Gaans, Stochastic integration for Levy processes with values in Banach spaces,
*Stochastic Process. Appl.*, to appear. (manuscript pdf file). - Bas Lemmens and Onno van Gaans, Dynamics of nonexpansive maps on strictly convex Banach spaces, to appear in:
*Israel J. Math*(manuscript pdf file). - Onno van Gaans and Anke Kalauch, Ideals and bands in pre-Riesz spaces,
*Positivity***12**(2008), no. 4, 591-611. (manuscript pdf file) - O. van Gaans and A. Kalauch, Bands in pervasive pre-Riesz spaces,
*Oper. Matrices***2**(2008), no. 2, 177-191. (manuscript pdf file) - M. Reiss, M. Riedle and O. van Gaans, On Emery's inequality and a variation-of-constants-formula,
*Stoch. Anal. Appl.***25**(2007), 353-379. (manuscript ps file, pdf file)

- (with A. Kalauch) Ideals and Bands in Pre-Riesz Spaces, report MATH-AN-01-2006, Institute of Analysis, TU Dresden, 2006. pdf file