Transient solution of a random walk with chemical rule

Conolly et al. [Math. Scientist 22 (1997) 83–91] have obtained the transient distribution for a random walk moving on the integers -∞<k<∞ of the real line. Their analysis is based on a generating function technique. In this paper, an alternative technique is used to derive elegant explicit expressions for the transient state distribution of an infinite random walk having “chemical” rule and starting initially at any arbitrary integer position (say i). As a special case of our result, Conolly et al.'s (1997) solution is easily obtained. Moreover, the transient solution of the infinite symmetric continuous random walk is also presented. Finally, numerical values testing the quality of our analytical results are illustrated.