Molecular diffusion from an instantaneous point in a laminar, two-dimensional flow field is studied by a random walk simulation performed over an approximately thousand-fold range of Peclet numbers. Isopleths of the diffusing cloud were obtained. An analysis of the statistical parameters of the cloud, both in the transversal and longitudinal direction, shows the aspects of the process that can be described analytically, as well as the significance of the concept of memory of the source. The methodology of the simulation, the relationship to the continuum approach, and the applications to other situations are examined.