Re: When agents cross paths?

[gross]

Paul,
    The questions you ask have been been long addressed in the random walk
literature - the general area is called recurrence time and relies on
the theory of stopping times for Markov processes. For the two dimensional case,
you might look at Frank Spitzer's classic text "Principles of Random Walk"
which focuses entirely on problems on lattices. This is also closely
linked to Potential Theory and harmonic analysis - for the associated 
problems in continuous space. The general area of diffusion processes, dealing
as it does with extensions of Brownian motion processes, includes many results
on related issues in process paths which cross - the classic text is Ito 
and McKean, "Diffusion Processes and their Sample Spaces", but a more readable 
suggestion is Karlin and Taylor's "Second Course on Stochastic Processes". 
    Note that much of the above literature is highly technical and difficult to
follow without a decent probability background. For anything other than the 
simplest individual agent models, the results are at best subjective - this is
because realistic individual agent models are typically highly complex, 
non-Markov, with state-dependent transition probabilities. The stochastic 
process literature deals mostly with analytically soluble problems.
    Cheers, 
        Lou Gross
Professor of Ecology and Evolutionary Biology
    and Mathematics
The Institute for Environmental Modeling
University of Tennessee - Knoxville
address@hidden
http://www.tiem.utk.edu/~gross/
http://archives.math.utk.edu/mathbio/ (Math Archives for Life Sciences)


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