What is the probability space corresponding to the probability measure $\mathbb{P}_{p}$ in the context of this paper?

Question

Here is the definition of the frog model we are interested in:

"... consider the homogeneous tree $\mathbb{T}_{d}$, that is, the rooted tree in which each vertex has (is connected by edges to) $d + 1$ neighbours. One frog is put at each vertex and all but the one of the root start inactive. Active frogs perform simple symmetric random walks on $\mathbb{T}_{d}$, for a geometric (parameter $1 - p$) number of steps, activating the inactive frogs of the visited vertices. After its geometric number of steps, the active frog “dies”: it remains inactive forever. The process survives if infinitely many frogs are activated.

For any $p\in[0,1]$, we denote the law of the process by $\mathbb{P}_{p}$. Naturally, if $p = 0$ then the frog of the root dies and $\mathbb{P}_{p}(\text{survival}) = 0$, while on the other hand, if $p = 1$, frogs won't die and $\mathbb{P}_{p}(\text{survival}) = 1$. Moreover, it is clear that $\mathbb{P}_{p}(\text{survival})$ is non-decreasing in $p$, and we can define the critical parameter for the model on $\mathbb{T}_{d}$ as \begin{align*} p_{c} = p_{c}(d) := \inf\{p\in[0,1] : \mathbb{P}_{p}(\text{survival}) > 0\}." \end{align*}

My questions are: what is the probability space associated to the probability measure $\mathbb{P}_{p}$? How can one describe the event "survival" rigorously? How do we prove that $\mathbb{P}_{p}(\text{survival})$ is non-decreasing in $p$?

I am talking about this paper.

Any references are welcome.

Answer 1

(1) As you read the probability literature, you'll soon discover that people tend not to be very specific about the sample space that they are working with. The details of that space are not important; what matters is the collection of observables (events and random variables) and their joint distribution. Any probability space that supports such a distribution is then fine. So the $\mathcal{F}$ and $\mathbb{P}$ of the probability space $(\Omega, \mathcal{F}, \mathbb{P})$ get much more attention than the $\Omega$. Here for example, it's enough that you have countably many simple symmetric random walks (one for each site of the tree), and countably many Geometric($p$) random variables, all of these independent.

[Sometimes some "canonical" choice of space might suggest itself; e.g. if you were studying bond percolation on a graph with edge set $E$, then you might explicitly take $\Omega=\{0,1\}^E$; but even then it's basically a matter of taste.]

(2) In the particular case of the frog model you describe, starting from a single active frog, one can see that in any bounded time-interval, there will be only finitely many jumps. Consequently there's actually a straightforward way to set up the model as a continuous-time Markov chain on a countable state space. The event of "survival" is then just the event that this Markov chain avoids some given state for ever.

(3) You might like to consider more general frog models, e.g. where an infinite number of frogs might be alive at the same time. Now the state space becomes uncountable, and the times of jumps may become dense, and things become a bit more subtle. However, I don't think there are any particular difficulties about the frog model beyond what you already need to construct more well-known interacting particle systems such as exclusion processes, contact processes, voter models, etc.

For a fairly rigorous account of the construction of such models, the books of Liggett might suit you well, e.g. Stochastic Interacting Systems. There are a ton of other resources introducing interacting particle systems (and many people who work with them, e.g. from a biology or physics viewpoint, are comfortable with a more informal approach to the construction).

(4) Monotonicity properties for interacting particle systems, like the one you mention, can be proved in various ways. A frequent approach involves coupling; you couple one copy of the process with parameter $p$ and another copy with parameter $p'>p$ in such a way that the path of the $p'$-process always lies above that of the $p$-process in some suitable sense. Again, if you read up about exclusion processes, contact processes, etc, you will find lots of examples of this sort of argument!