Some results on first passage times in one dimensional random walks

Analytic expressions are presented for the characteristic function of the first passage time distribution for biased random walk on a finite chain (and diffusion with drift on a finite line); of the first passage time distribution for a random walk on a chain, in which the events (jumps) are governed by an arbitrary renewal process; and of the distribution of the time of escape from a bounded set of points in the latter case. A fundamental relation between the first passage time distribution and the conditional probability for random walk (or diffusion) in one dimension is analyzed and generalized.