[jason1990]

I was preparing another long post with some new conclusions when I saw this. So I will first reply here and then post what I was working on.

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It seems like you put a lot of thought into this. I think I agree with most or everything you write, except I think it would be strange to select a non-uniform u.

First of all, if u is not uniform, and the observer starts looking at negative t, then he will find that the fraction of +2 jumps will converge (in the limit of many observations of negative t) towards something else than 0.5. How is this consistent with the problem?

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First, let me point out that there are 2 aspects of the problem with which we would like to be consistent: (1) the frog jumps forward in time by +2 or -1 with equal probability, and (2) the frog came from -infty.

Now, the questions that have interested me here are these. How can we rigorously construct the frog's movement backward in time? Is a particular such construction forced on us by either (1) alone or by (1) and (2) together?

I am satisfied by the Poisson cloud construction. We can construct this cloud as a legitimate (Markov) stochastic process taking values in the space of point configurations, i.e. the set of functions from the integers to the nonnegative integers. We can construct it for all time, both positive and negative, by making it have a stationary distribution. The stationary distributions are the ones that have a Poisson number of particles at each site with parameter u(i), and with each site being independent. We can then consider tracer particles in this system if we want to consider individual frogs.

I personally do not find it strange to use a non-uniform u. All stationary distributions for the cloud seem equally legitimate to me. With a non-uniform u, a tracer particle will jump +2 and -1 with equal probability in the forward time direction, but when we look back in time the proportion of +2 jumps does not converge to 0.5, as you pointed out. I find this surprising and interesting, but it is definitely consistent with (1). "The math speaks for itself," as PairTheBoard said.

However, are any of these non-uniform distributions consistent with (1) and (2) together? This I will address in my next post.