The motion of a particle released into a fluid flow depends upon the velocity of the fluid in the immediate vicinity of the particle. In a turbulent flow, the fluid velocity is a random quantity, so that particle motions are also random. Knowledge of certain statistical characteristics of the fluid motion allows for determination of mean properties of the particle motions. In this paper, dispersion of particles is analysed by assuming a simple form for the auto-correlation of the fluid motions following the particle in homogeneous, isotropic and stationary turbulence. Two cases are considered, one where the particles are initially at rest, the second where the particles are initially ’excited’. Particle velocity auto-correlations, particle dispersion coefficients and related quantities are determined as functions of time for both cases. Expressions for development times for heavy particles are derived.