[jason1990]

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Looking at the measure with parameter k,

u(i) = 1 + kc^i

with a Large Scale view, the graph looks the same regardless of how small you choose k. If you make k smaller, just step back and take an even Larger Scale view of it. On the right you're going to see a line with an extremely high positive slope. Translating this to the Poison stacks of frogs you descibe below for the Cloud of Frogs you might see had they all began at -infinity,

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They did not all necessarily begin at -infinity. More on this below.

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with the Large Scale View the Fuzziness nearly disappears. And to the right where you see an extremely high slope to the line 1+kc^i, it sure seems like you are going to see a lot more frogs jumping -1 back to i than +2 forward to it.

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This confuses me, since "back" and "forward" can refer to time and/or space.

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Thus, still producing a positive backward drift. Although, on the left where 1+kc^i locally looks uniform, you would not expect to see the positive backward drift. It's hard to believe that produces a consistent local q(i,j).

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Okay, this is a great point. First of all, you are using the word "local," but I think you really mean "homogeneous" or "translation invariant." A time-homogeneous Markov process has a transition function q(i,j) which does not depend on t. It may also be the case that q depends only on j-i. This implies, for example, that the probability of jumping from i to i+1 does not depend on i. You might want to call this "spatially homogeneous." But it may be better just to call this a process with iid increments.

Only in the cases k_1 = 0 or k_2 = 0 do you get a reverse-time process which has iid increments. For other choices, the reverse-time transition function depends on both i and the jump size. (This, however, does not necessarily mean that the origin plays any special role. If you redefine your origin to be i_0, you simply need to redefine your k_2 to be k_2 c^{i_0}.)

For u(i) = 1 + kc^i, we get

q(i,i+1) = (1 + kc^{i+1})/(2 + 2kc^i), and
q(i,i-2) = (1 + kc^{i-2})/(2 + 2kc^i).

Now, every single frog in that Poisson cloud is going to be moving forward in time according the rules in the original puzzle. But they will be moving backward in time according the transition function above, which depends on i. Those frogs that we presently see in a very dense part of the cloud will move backward in time with an immediate positive spatial drift. This ought to carry them (back in time) farther into the denseness of the cloud, and their drift ought to increase toward some finite maximum. So those frogs ought to have "come from +infinity." Of course, there is some probability that they will initially move against their mean drift. In that case, there will be some "reinforcement," since by moving in the negative space direction, they will decrease their mean drift, making it even easier for them to continue moving against their mean drift. So it is not clear to me right now where these frogs came from. I would still expect that the frogs in the very dense parts of the cloud came from +infinity.

We can do the same analysis with the frogs that we presently see in a very sparse part of the cloud. They will move backward in time with an immediate negative spatial drift. This ought to carry them (back in time) farther into the sparseness of the cloud, and their drift ought to decrease toward the finite minimum -1/2. So those frogs ought to have "come from -infinity." But again there is the possibility that, despite their mean negative drift, they move (back in time) in the positive spatial direction, which changes their drift and helps them to possibly continue moving in this direction. Again, I expect the frogs in the very sparse parts of the cloud to have come from -infinity. This actually looks very interesting.

This cloud model is an example of an evolving point process. When we ask about the motion of a particular particle in the point process, we are asking about a so-called "tracer particle." These can be complicated and interesting models. In this case, the cloud itself remains stationary. The frogs in the cloud all behave identically in the forward time direction. But their behaviors are different in the backward time direction.

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I may respond more to the rest of your post(s) in a while. However, I'd like to hear what you think the implications are of this idea.

Suppose instead of conditioning on where we see the frog at our initial reference time, We instead condition on the First Time the Frog enters the interval [0,1].

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I do not think I will have time to chew on this for a while. But you may ask yourself what it is you want to condition. Is it the nonprocess S defined on the measure space with infinite mass? Is it a particular frog in the Poisson cloud? If the latter, which frog? I assume you would want to choose a frog that came in from -infinity, but there may be many of them. Although they all have the same behavior in the forward time direction, they do not all have the same behavior in the backward time direction.