Geodesic
Crossing an Asymptotically Square-Root Boundary by the Brownian Motion
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 113-128

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We consider first-passage times of the standard Brownian motion over boundaries of order $c\sqrt {t}$. Our main result determines the tail behaviour of such first-passage times. This generalizes the well-known results obtained by Novikov and Uchiyama.
Keywords: Brownian motion, space-time harmonic function, parabolic cylinder function. 
@article{TM_2022_316_a8,
     author = {Denis E. Denisov and G\"unter Hinrichs and Alexander I. Sakhanenko and Vitali I. Wachtel},
     title = {Crossing an {Asymptotically} {Square-Root} {Boundary} by the {Brownian} {Motion}},
     journal = {Trudy Matematicheskogo Instituta imeni V.A. Steklova},
     pages = {113--128},
     publisher = {mathdoc},
     volume = {316},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TM_2022_316_a8/}
}
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Denis E. Denisov; Günter Hinrichs; Alexander I. Sakhanenko; Vitali I. Wachtel. Crossing an Asymptotically Square-Root Boundary by the Brownian Motion. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Branching Processes and Related Topics, Tome 316 (2022), pp. 113-128. http://geodesic.mathdoc.fr/item/TM_2022_316_a8/