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. 
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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/

[1] Breiman L., “First exit times from a square root boundary”, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, Pt. 2: Contributions to probability theory (Univ. Calif. 1965/1966), v. II, Univ. Calif. Press, Berkeley, 1967, 9–16 | MR | Zbl

[2] Denisov D., Sakhanenko A., Wachtel V., “First-passage times for random walks with nonidentically distributed increments”, Ann. Probab., 46:6 (2018), 3313–3350 | DOI | MR | Zbl

[3] D. Denisov, A. Sakhanenko, and V. Wachtel, “First-passage times over moving boundaries for asymptotically stable walks”, Theory Probab. Appl., 63:4 (2019), 613–633 | DOI | MR | Zbl

[4] Gärtner J., “Location of wave fronts for the multi-dimensional K-P-P equation and Brownian first exit densities”, Math. Nachr., 105 (1982), 317–351 | DOI | MR | Zbl

[5] A. A. Novikov, “On stopping times for a Wiener process”, Theory Probab. Appl., 16:3 (1971), 449–456 | DOI | MR | Zbl

[6] A. A. Novikov, “On estimates and the asymptotic behavior of nonexit probabilities of a Wiener process to a moving boundary”, Math. USSR, Sb., 38:4 (1981), 495–505 | DOI | MR | Zbl | Zbl

[7] A. A. Novikov, “A martingale approach in problems on first crossing time of nonlinear boundaries”, Proc. Steklov Inst. Math., 158 (1983), 141–163 | MR | Zbl | Zbl

[8] A. A. Novikov, “Martingales, Tauberian theorem, and strategies of gambling”, Theory Probab. Appl., 41:4 (1997), 716–729 | DOI | MR | MR | Zbl

[9] Oldham K.B., Myland J., Spanier J., An atlas of functions, 2nd ed., Springer, New York, 2009 | MR

[10] Sato S., “Evaluation of the first-passage time probability to a square root boundary for the Wiener process”, J. Appl. Probab., 14 (1977), 850–856 | DOI | MR | Zbl

[11] Shepp L.A., “A first passage problem for the Wiener process”, Ann. Math. Stat., 38 (1967), 1912–1914 | DOI | MR | Zbl

[12] Taksar M.I., “First hitting time of curvilinear boundary by Wiener process”, Ann. Probab., 10 (1982), 1029–1031 | DOI | MR | Zbl

[13] Uchiyama K., “Brownian first exit from and sojourn over one sided moving boundary and application”, Z. Wahrscheinlichkeitstheor. verw. Geb., 54 (1980), 75–116 | DOI | MR | Zbl