Second Workshop on
Fractional Calculus, Probability and Non-Local Operators: Applications and Recent Developments
BCAM, Bilbao, Basque Country - Spain

November 10-14 2014

Research on non-local operators and related stochastic processes has witnessed a rapid expansion over the past years. These problems pose new challenges to both pure and applied mathematicians, at the interface of at least three wide fields: functional analysis, stochastic processes and partial differential equations, while providing a new paradigm in scientific modelling. Applications range from relativistic models of quantum physics and quantum field theory, to sub-diffusion experimentally observed in molecular physics, from anomalous transport in cells to leptokurtic price distributions in financial markets, or to long-range dynamics ranging from the nano-scale to the geophysical scale. This interaction between mathematics and applications is giving rise to new concepts and methods, and it will produce new challenging mathematical problems for many years to come.

Program

Monday 10th - Wednesday 12th
Mon 11:30-13:30 / 15:00-19:00
Tue 9:00-12:00 / 15:00-16:00 / 17:00-19:00
Wed 9:00-12:00 / 15:00–18:00

PhD/PostDoc level course on "Fractional Kinetics: Analytic and Probabilistic Approaches"

Vassili Kolokoltsov (University of Warwick, UK)
Jozsef Lorinczi (Loughborough University, UK)
Enrico Scalas (University of Sussex, UK)

Aims:

The aim of the course is the presentation in a didactic way of results concerning some issues that strongly relates stochastic processes and fractional calculus.
The course will start with an introduction of the methods of the theory of Markov processes and asymptotic analysis for the study of the scaling limits of CTRW (continuous time random walks)and related fractional, in space and time, differential equations.

Further small Lévy-noise perturbations of dynamical systems will be analysed. In particularin the lectures it will be considered how the trajectories of a differentiable dynamical system given by an ODE behave under a perturbation by an additive noise term given by a Lévy process.

In the so obtained Lévy-driven SDE the coupling constant of the noise will be assumed to be small. A main question considered is what are mean exit times and typical exit paths the perturbed system is likely to follow if the initial point is inside a given domain containing a stable attractor of the original ODE. The lectures will proceed in a comparative way by first choosing the perturbation to be Gaussian (i.e., Brownian motion). Next it will be chosen to be a non-Gaussian stable process, and finally it will be discussed how the exit patterns and mechanisms differ in function of the heavy/light tail properties of the perturbing noise.

If time allows, it will be also discussed related metastability phenomena.

Finally, the fractional Poisson process and its applications are discussed. The fractional Poisson process is a counting renewal process that was independently introduced in the first half of the previous decade by various authors and has recently been further characterised. It generalises the Poisson process and is naturally related to time-fractional diffusion. It finds applications in several applied fields including biophysics, finance and meteorology.

Program:

Markov processes, semigroups and generators.
Convergence of Markov processes.
Convergence of scaled CTRW and their position dependent analogs.
Controlled CTRW and their scaling limits.
Stable Lévy motions, stable-like processes and fractionally stable laws.
The theory of Fractional Hamilton-Jacobi-Bellmann equations.
Large deviation technique.
Definition of the fractional Poisson process. Properties of the fractional Poisson process.
The compound fractional Poisson process. Relationship with time-fractional diffusion. Applications



Thursday 13th - Friday 14th
Thu, Fri 9:30-12:00 / 15:00-17:30

Workshop with specialist talks presented by Invited Speakers and by young researchers. A permanent poster session will be organised as well.

Invited Talks:

Luisa Beghin (University "La Sapienza" Rome, Italy) ”Stable processes at Poisson times in connection with the fractional shift operator"

Kamil Kaleta (University of Warsaw, Poland) “Lévy processes with uniformly dominated sequences of jumps"

Joseba Mendiguren (Mondragon University, Spain) “Fractional Calculus application on mechanical engineering problems"

Paolo Paradisi (ISTI-CNR Pisa, Italy/BCAM, Spain) “Stochastic processes for space-time fractional diffusion”

Juan Luis Vázquez (Autonomous University of Madrid, Spain) “Nonlinear diffusion with fractional Laplacian operator”



Grants are available for attendees.
For registration write to fcpnlo@gmail.com, no registration fee is required.

To submit your abstract for a poster or a presentation (deadline: 24th of October, 2014) and further information, please write to: fcpnlo@gmail.com

Sincerely Yours,
Jozsef Lorinczi
Gianni Pagnini
Enrico Scalas