Diffusion Processes with a Weakly Absorbing Boundary

The paper discusses homogeneous diffusion processes on $[{0,\infty})$ whose behavior on the boundary $x=0$ is determined by condition (1), where the parameter $\varepsilon$ characterizes the intensity of absorption of the process on the boundary. The limiting behavior of the distribution of the duration of the process, as $\varepsilon \to 0$, is examined.