In the book "A guide to First-Passage Processes" by Sidney Redner, a section is dedicated to the survival probability of a random walker in a growing domain.
For a fixed-length interval $[0,L]$, the probability that a diffusing particle has remained in the interval asymptotically decays as $e^{-\pi^2Dt/L^2}$ for large times $t$. The chapter considers cases where the interval expands over time: $L(t)\propto t^{\alpha}$.
However, the book only looks at continuous space situations. I would be interested to know some of the literature that looks at (continuous time) discrete-space (CTRW) situations. After a brief search, starting from the paper of Redner, I could not find anything.
Is there any work that look at CTRW in 1d on an expanding domain? (of course there is..., but who,where?)
I would be interested to know about e.g. the probability that a walker starting at site $i$ on a lattice of size $N(t)$ reaches either extremities of the lattice where the size of the lattice expands with time.
