I have a discrete time stochastic process where an interval from 0 to L consists of smaller sub-segments, where the boundaries are always at integers, and L is some large integer. At each iteration, a segment is chosen with probability proportional to its length, and an operation is performed on it, such as growing it while shrinking the neighboring segment, shrinking it while growing the neighboring segment, or cutting it into two smaller segments. The quantitative details are not important here.

Here's a sample realization of the process

What analytic approach can I use for creating probabilities of the positions of boundaries over time? It seems difficult since the number of boundaries can change. I am familiar with Markov chains, but I'm not sure what the state space would be. I can certainly model this numerically, but I'm looking for a theoretical approach.

Even if you don't know that answer, saying "this is similar to a XX process" would be great.

  • $\begingroup$ It is difficult to see how one could possibly answer the question "how can I model" without being told the "exact details"! What are the rules by which the operations are chosen and carried out and what do they mean? For instance, can one segment be "grown" without changing the lengths of others? If so, how do the others move to accommodate the growth? $\endgroup$ – whuber May 1 '20 at 20:24
  • $\begingroup$ Okay, I added more details. I wanted to keep it a little vague because I'm just interested in an overall approach, and don't expect a precise answer. $\endgroup$ – user3433489 May 1 '20 at 20:32

A similar problem is the Ehrenfest urn model. In the simplest case, there are 2 urns, and M balls distributed among them. At each time, a ball is randomly chosen with probability 1/M, and is placed into the opposite urn. The situation in the original post is related to an urn model with N urns. The N-urn model is more complicated, but has been discussed in the academic literature. The issue of new intervals being created would correspond to to new urns being added during the process. I haven't seen anyone discuss that.


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