Fractional Brownian motion with time- and space-dependent Hurst exponent


Friday, 19 July, 2024 - 15:00 to 16:00

Speaker : Ralf Metzler, Institute of Physics & Astronomy, University of Potsdam, 14476 Potsdam, Germany

Time : 15:00 - 16.00 CEST (Rome/Paris)

Hosted at: SISSA, International School of Advanced Studies, Trieste, Italy

Zoom : A link will appear here, an hour before the talk.

Organizers : Pavan Pranjivan Mehta* ( and Arran Fernandez** (

* SISSA, International School of Advanced Studies, Italy

** Eastern Mediterranean University, Northern Cyprus

Keywords: Anamolous diffusion, Long-range dependence, Fractional Brownian motion

Abstract: Stochastic processes with long-range dependent correlations naturally emerge in many systems when all degrees of freedom are integrated out, apart from the (tracer) particle of interest. Examples are the dynamics of a labelled monomer in Rouse-type polymer chains or test particles in viscoelastic solutions. In non-equilibrium situations, the resulting overdamped dynamics corresponds to fractional Brownian motion (FBM). This talk will first address disordered systems, when the observed displacement probability density turns out to be non-Gaussian. For the descriotion of such situations a random-diffusivity (doubly-stochastic) FBM model [1, 2] will be introduced and shown that the precise dynamics encoded in this model delicately depends on its precise mathematical formulation.

This talk will then introduce Lévy’s formulation of FBM via a Riemann-Liouville fractional integral. It will be discussed how time- and space-dependent scaling (Hurst) exponents and diffusion coefficients can be introduced in this model, and which dynamic behaviour emerges [3, 4]. Again, the specific formulation of the model results in significantly different dynamical behaviours.

Biography: After receiving his doctorate at the University of Ulm, Ralf Metzler went to Tel Aviv University for a postdoc with Prof Yossi Klafter. He then continued for a second postdoc with Prof Mehran Kardar at MIT before his first faculty appointment as assistant professor at the Nordic Institute for Theoretical Physics (NORDITA) in Copenhagen. Ralf then became associate professor and Canada Resarch Chair for Biological Physics at the University of Ottawa. In 2007 Ralf was appointed as professor at the Technical University of Munich, and since 2011 he holds the chair for Theoretical Physics at the University of Potsdam. In 2010 Ralf was also a Finland Distinguished Professor at Tampere University of Technology. Among others, Ralf was awarded the 2017 SigmaPhi prize and an Alexander von Humboldt Polish Honorary Research Scholarship (2018-2022). Ralf’s work focuses on statistical and biological physics, in particular, on stochastic processes and their applications.


[1] A. V. Chechkin, F. Seno, R. Metzler, and I. M. Sokolov. Brownian yet non-Gaussian diffusion: from superstatistics to subordination of diffusing diffusivities. Phys. Rev. X 7, 021002 (2017).

[2] W. Wang, F. Seno, I. M. Sokolov, A. V. Chechkin, and R. Metzler Unexpected crossovers in correlated random-diffusivity processes New J. Phys. 22, 083041 (2020).
W. Wang, A. G. Cherstvy, A. V. Chechkin, S. Thapa, F. Seno, X. Liu, and R. Metzler. Fractional Brownian motion with random diffusivity: emerging residual nonergodicity below the correlation time. J. Phys. A 53, 474001 (2020).

[3] W. Wang, M. Balcerek, K. Burnecki, A. V. Chechkin, S. Janušonis, J. Ślęzak, T. Vojta, A. Wyłomańska, and R. Metzler. Memory-multi-fractional Brownian motion with continuous correlations. Phys. Rev. Res. 5, L032025 (2023).

[4] J. Ślęzak and R. Metzler. Minimal model of diffusion with time changing Hurst exponent. J. Phys. A 56, 35LT01 (2023)