First-Passage Time

We consider the first-passage time problem for Brownian motion on a line. We derive an expression for the PDF of the time when the diffusing particle, starting from some arbitrary initial point x0 at \( {\text{t}} = 0 \), reaches any other given point x for the first time. Two different methods are used for this purpose. The first relies on a renewal equation for Brownian motion. The second determines the PDF required from a certain survival probability. The PDF is shown to correspond to a stable distribution with characteristic exponent \( \frac{1}{2} \). The mean and higher moments of this distribution are infinite. We discuss the nature of this divergence. The distribution of the time to reach a given distance from the starting point for the first time is also determined.