For normally reflected Brownian motion and for simple random walk on independently growing in time $d$-dimensional domains, $d\ge3$, we establish a sharp criterion for recurrence versus transience in terms of the growth rate.
"Walking within growing domains: recurrence versus transience." Electron. J. Probab. 19 1 - 20, 2014. https://doi.org/10.1214/EJP.v19-3272