This process contains the Walsh Brownian motion as a special case, which was first considered in the epilogue of Walsh ( 1978), and further studied by Rogers ( 1983), Baxter and Chacon ( 1982), Salisbury ( 1986) and Barlow et al. ( 2012) and Fitzsimmons and Kuter ( 2015). The construction of Brownian motion on general metric graphs can be seen in Georgakopoulos and Kolesko ( 2014), Kostrykin et al. In this paper, we are interested in the Brownian motion moving on a simple graph. ( 2014) constructed a Volterra integral system for the probability density function of the first hitting time of Brownian motion with a general two-sided boundary. Che and Dassios ( 2013) used a martingale method to derive the crossing probability with a two-sided boundary involving random jumps. Escribá ( 1987) studied the crossing problem with two sloping line boundaries. Peskir ( 2002) provided a general result for the continuous boundary using the Chapman-Kolmogorov formula, and gave the probability density function of the first hitting time in terms of a Volterra integral system.įor the first hitting time of Brownian motion with a two-sided boundary, the Laplace transform and density are well-known, see Borodin and Salminen ( 1996) Section II.1.3. Scheike ( 1992) derived an exact formula for the broken linear boundary. Wang and Pötzelberger ( 1997) obtained the crossing probability for Brownian motion with a piecewise linear boundary using the Brownian bridge. The square-root boundary was considered by Breiman ( 1967) via the Doob’s transform approach. The second-order boundary was studied by Salminen ( 1988) using the infinitesimal generator method. Other types of boundary have also been considered. The study of first hitting time of Brownian motion with linear boundary goes back to Doob ( 1949).
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