The First Exit Time of a Brownian Motion from the Minimum and Maximum Parabolic Domains

Dawei Lu () and Lixin Song ()
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Dawei Lu: Dalian University of Technology
Lixin Song: Dalian University of Technology

Journal of Theoretical Probability, 2011, vol. 24, issue 4, 1028-1043

Abstract: Abstract Consider a Brownian motion starting at an interior point of the minimum or maximum parabolic domains, namely, $D_{\min}=\{(x,y_{1},y_{2}):\|x\| 1. Let $\tau_{D_{\min}}$ and $\tau_{D_{\max}}$ denote the first times the Brownian motion exits from D min and D max . Estimates with exact constants for the asymptotics of $\log P(\tau_{D_{\min}}>t)$ and $\log P(\tau_{D_{\max}}>t)$ are given as t→∞, depending on the relationship between p 1 and p 2, respectively. The proof methods are based on Gordon’s inequality and early works of Li, Lifshits, and Shi in the single general parabolic domain case.

Keywords: Brownian motion; Bessel process; Gordon’s inequality; Exit probabilities; 60G15; 60G40; 60F10 (search for similar items in EconPapers)
Date: 2011
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10959-010-0306-7

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