In the preprint [3], Goldston, Pintz, and Yıldırım established, among other things, \begin{equation} \liminf_{n\to\infty}{p_{n+1}-p_n\over\log p_n}=0, \end{equation} with $p_n$ the $n$th prime. In the present article, which is essentially self-contained, we shall develop a simplified account of the method used in [3]. We include a short expository last section.

The purpose of this paper is to study the collision or non-collision problem for interacting Brownian particles in the framework of theory of Dirichlet forms. The result is closely related to a question on existence and uniqueness of strong solutions for stochastic differential equations with singular drifts.