Hitting Spheres with Brownian Motion and Sommerfeld′s Radiation Condition

The joint distribution of the hitting ti

The joint distribution of the hitting time and place to a sphere or spherical shell for Brownian motion with drift

✍ Chuancun Yin 📂 Article 📅 1999 🏛 Elsevier Science 🌐 English ⚖ 95 KB

Let Xt be a standard d-dimensional Brownian motion with drift c started at a ÿxed X0, and let T be the hitting time for a sphere or concentric spherical shell. By using an appropriate martingale, a Laplace-Gegenbauer transform of the joint distribution of T and XT is determined.