Two-scale renormalization-group classification of diffusive processes.

Renormalization-group operators are used to classify stochastic processes on two time scales. Repeated application of one operator is associated with the long-time behavior of the process while the other is associated with the short-time behavior of the process. This approach is shown to be robust even in the presence of nonstationary increments and infinite second moments. Fixed points of the operators can be used for further subclassification of processes when appropriate limits exist. Several processes are classified using the renormalization-group scheme. The processes to be classified include advection-diffusion in an ergodic velocity field, and a model of diffusion in the human bronchial tree.

[1]  D. Panja.  Probabilistic phase space trajectory description for anomalous polymer dynamics , 2010, Journal of physics. Condensed matter : an Institute of Physics journal.

[2]  John H. Cushman,et al.  A Renormalization Group Classification of Nonstationary and/or Infinite Second Moment Diffusive Processes , 2012 .

[3]  J. Klafter,et al.  Fractional brownian motion versus the continuous-time random walk: a simple test for subdiffusive dynamics. , 2009, Physical review letters.

[4]  J. H. Cushman,et al.  A stochastic model for anomalous diffusion in confined nano-films near a strain-induced critical point , 2011 .

[5]  Mehran Kardar,et al.  Anomalous diffusion with absorbing boundary. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  R. Metzler,et al.  In vivo anomalous diffusion and weak ergodicity breaking of lipid granules. , 2010, Physical review letters.

[7]  Diogo Bolster,et al.  Anomalous mixing and reaction induced by superdiffusive nonlocal transport. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[8]  Fractional brownian motion run with a nonlinear clock. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  J. H. Cushman,et al.  Renormalizing chaotic dynamics in fractal porous media with application to microbe motility , 2006 .

[10]  Nicolas E. Humphries,et al.  Environmental context explains Lévy and Brownian movement patterns of marine predators , 2010, Nature.

[11]  C. Gardiner.  Handbook of Stochastic Methods , 1983 .

[12]  J. H. Cushman,et al.  Anomalous diffusion as modeled by a nonstationary extension of Brownian motion. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Chaotic dynamics of super‐diffusion revisited , 2009 .

[14]  J. H. Cushman,et al.  Scaling laws for fractional Brownian motion with power-law clock , 2011 .

[15]  Fractional Brownian sheets run with nonlinear clocks , 2012 .

[16]  D. Benson,et al.  Multidimensional advection and fractional dispersion. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .