The distribution of particles in the plane dispersed by a simple 3-dimensional diffusion process

Populations of particles dispersed in the 2-dimensional plane from a single point-source may be grouped as focus expansion patterns, with an exponentially decreasing density, and more diffuse patterns with thicker tails. Exponentially decreasing distributions are often modelled as the result of 2-dimensional diffusion processes acting to disperse the particles, while thick-tailed distributions tend to be modelled by purely descriptive distributions. Models based on the Cauchy distribution have been suggested, but these have not been related to diffusion modelling. However, the distribution of particles dispersed from a point source by a 3-dimensional Brownian motion that incorporates a constant drift, under the condition that the particle starts at a given height and is stopped when it reaches the xy plane (zero height) may be shown to result in both slim-tailed exponentially decreasing densities, and thick-tailed polynomially decreasing densities with infinite mean travel distance from the source, depending on parameter values. The drift in the third coordinate represents gravitation, while the drift in the first and second represents a (constant) wind. Conditions for the density having exponentially decreasing tails is derived in terms of gravitation and wind, with a special emphasis on applications to light-weighted particles such as fungal spores.