Explaining Continuous Time Random Walks: What Is It and How Is It Used?

[yaboidjaf]

Hi, I'm trying to read into CTRW, but I'm finding the information online a little difficult to take in. From what I've read the process differs from normal random walks in that jumps take place after some waiting time $$\tau$$, which can be from 0<$$\tau$$<$$\infty$$. Would I also be right in saying that the jump distance is continuous as well?

After that I come to the Montroll-Weiss equation, Fourier and Laplace transforms and work with probabilities about the positioning after a certain number of steps or time. I'm really looking for something that explains the meaning of all the maths, what the probabilities and transforms show about a particle undergoing this process. Maybe a practical application or a simpler explanation even? Any suggested sources?

Any help would be much appreciated. Thanks.

 

[D H]

Hmm. My initial thought when I read your post was "Yikes! Yet another name for a Gauss-Markov process"! But you are talking about something else.

I found this paper; if you've already come across it, never mind!

Meerschaert et al, Governing equations and solutions of anomalous random walk limits, PRevE, 66:060102
http://inside.mines.edu/~dbenson/current/couplePRE.pdf

 

[Andy Resnick]

AFAICT, the main motivation for developing this line of study is to model "anomalous diffusive transport". That is, diffusive processes where the mean squared displacement <x^2> != t, but is instead a power law t^a.

This situation occurs for diffusion in disordered media:
http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6TVP-46SXPMN-7F&_user=7774119&_coverDate=11%2F30%2F1990&_rdoc=1&_fmt=high&_orig=search&_origin=search&_sort=d&_docanchor=&view=c&_acct=C000062847&_version=1&_urlVersion=0&_userid=7774119&md5=7f2d1c134653937175696ea89d5db97d&searchtype=a

cell migration:
http://www.mpipks-dresden.mpg.de/~rklages/publ/publ.html[/URL]

searching patterns of wandering animals:
[url]http://www.nature.com/nature/journal/v449/n7165/abs/nature06199.html[/url]

and intracellular transport:
[url]http://biophysics.physics.brown.edu/BPJC/JC%20pdf%20paper%20files/BPJC%20Fall%202005/CaspiPRL2000.pdf[/url]

 

[rkla]

Hi,
I found this discussion, because Andy linked to my publications (thanks!).
Perhaps Section 4 of a review is helpful that I wrote last year. It got published as a book chapter, http://arxiv.org/abs/0804.3068" .
It contains a number of further introductory references.
Good luck,
R.