Inspired somewhat by this problem, I've been investigating the behavior under iteration of the following discrete random process:
Given $n\in\mathbb{N}$, choose an integer from $\{0,1,\ldots,n\}$ uniformly at random, and multiply $n$ by it.
The process a.s. either reaches its fixed point, $0$, or diverges to infinity. (The only other possibility, that $n$ remains forever at some non-zero value, clearly has zero probability.) When it diverges to infinity, it does so quickly: its average value grows super-exponentially with the number of steps. I'm interested in the probability of the process reaching $0$, and in the cases where it does reach $0$, the largest value it attained.
Let $p(n)$ be the probability of reaching $0$ starting from $n$. Then we have $ p(n) = \frac{1}{n+1}\left(1 + \sum_{k=1}^n p(kn)\right), $ or $ p(n) = \frac{1}{n} + \frac{1}{n}\sum_{k=2}^n p(kn). $
We have $p(n) \ge 1/n$ immediately; feeding this back in gives an improved lower bound of $ p(n) \ge \frac{1}{n}\left(1 + \sum_{k=2}^{n}\frac{1}{kn}\right) = \frac{1}{n}+\frac{1}{n^2}\left(H_{n}-1\right), $ where $H_n$ is the $n$-th harmonic number. The lower bound can be improved by repeating this process; can a useful asymptotic expansion for $p(n)$, or even a closed form, be derived?
Similarly, let $q(n)$ be the expected value immediately preceding $0$ among terminating sequences that start from $n$. It is clear from the definition that $q(n)\ge n$, and numerical simulation suggests that $q(n) \sim K n$ for some constant $K$. Can $q(n)$ be given an asymptotic expansion? What is the value of the constant K?