There are several papers discussing the crossing of square root boundary by a random walk, which seem to be relevant.
- Breiman (1967). First exit times from a square root boundary. https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Fifth-Berkeley-Symposium-on-Mathematical-Statistics-and/chapter/First-exit-times-from-a-square-root-boundary/bsmsp/1200513456
Breiman (1967). First exit times from a square root boundary. https://projecteuclid.org/ebooks/berkeley-symposium-on-mathematical-statistics-and-probability/Proceedings-of-the-Fifth-Berkeley-Symposium-on-Mathematical-Statistics-and/chapter/First-exit-times-from-a-square-root-boundary/bsmsp/1200513456
- Greenwood and Perkins (1983). A conditioned limit theorem for a random walks and Brownian local time. https://projecteuclid.org/journals/annals-of-probability/volume-11/issue-2/A-Conditioned-Limit-Theorem-for-Random-Walk-and-Brownian-Local/10.1214/aop/1176993594.full
Greenwood and Perkins (1983). A conditioned limit theorem for a random walks and Brownian local time. https://projecteuclid.org/journals/annals-of-probability/volume-11/issue-2/A-Conditioned-Limit-Theorem-for-Random-Walk-and-Brownian-Local/10.1214/aop/1176993594.full
[Edit] Uchiyama (1980). Brownian first exit from and sojourn over one sided moving boundary and application. https://link.springer.com/article/10.1007/BF00535355