• Gaussian processes;
  • microrheology;
  • non-Newtonian fluids;
  • generalized Langevin equation;
  • local Whittle;
  • fractional processes;
  • simulation;
  • wavelets
  • Primary: 60G15, 82B31, 62P10;
  • Secondary: 62M09, 42C40

Microrheology is the study of the properties of a complex fluid through the diffusion dynamics of small particles, typically latex beads, moving through that material. Currently, it is the dominant technique in the study of the physical properties of biological fluids, of the material properties of membranes or the cytoplasm of cells, or of the entire cell. The theoretical underpinning of microrheology was given in Mason and Weitz (Physical Review Letters; 1995), who introduced a framework for the use of path data of diffusing particles to infer viscoelastic properties of its fluid environment. The multi-particle tracking techniques that were subsequently developed have presented numerous challenges for experimentalists and theoreticians. This study describes some specific challenges that await the attention of statisticians and applied probabilists. We describe relevant aspects of the physical theory, current inferential efforts and simulation aspects of a central model for the dynamics of nano-scale particles in viscoelastic fluids, the generalized Langevin equation.