Since the publications of Kolmogorov in the 1940s, the dominant approach has been to model turbulence using fractional or multi-fractional processes. The experimental confirmation of the predictions made possible by the models has always, however, been a delicate matter. Recently, the observation of time series taken from commodities or currency markets has made it possible to demonstrate in a direct way such multi-fractional behaviour.

The Cournot Centre’s Probabilism research programme for 2020 is organized around two questions stemming from these results:

  • On what kind of data is this type of behaviour observable?
  • Which micro-models are capable of taking into account these macroscopic processes?

The first results were based on data from human interactions. Can the methods developed for commodities or currencies be used on other types of data? Fields as varied as wave propagation in the atmosphere, the dynamics of animal populations, intracellular transport in systems biology, imaging of human tissue by elastography, communication networks and internet traffic are all areas in which the multi-fractional approach has been tested.

In all of these fields of study, the nature of the data conditions the development of the models to be estimated. They have to be sampled correctly, in order to allow for time-frequency analysis, and must not be filtered or pretreated. The fact that the processes are of a fractional or multi-fractional nature is fundamental. If modelling Markov processes has been the standard stochastic approach, the multiplication of available data confirms that it is not only possible, but necessary to go beyond this framework and develop new tools and non-Markovian models.

There are important challenges associated with modelling, simulation and analysis of non-Markovian phenomena when trying to understand them through comparison with experimental data. Non-Markovian models may, in particular, be important for modelling what results from human intervention; that may mean that there is “memory” in the process leading to a non-Markovian behaviour as, for instance, recently seen in price processes.