The Diffusion of Euclidean Shape

This paper is a preliminary report on the results of an investigation into the diffusion of Euclidean shape, using computer algebra to reduce complicated intermediate calculations to an informative final form. The computer algebra takes the form of an extension to the symbolic Itô calculus described in W.S. Kendall (1988). A substantially more detailed treatment (including details of how to obtain the results below and a description and discussion of the necessary extensions to symbolic Itô calculus) will be provided in a later paper. The results are new and will be of interest to workers in the field of statistics of shape, and perhaps also to mathematical physicists. They provide a reinforcement of the view expressed in W.S. Kendall (1988), and further argued in my contribution to the discussion of D.G. Kendall (1989), that computer algebra and the associated equipment now form a powerful tool for probabilists and statisticians, as indeed for the mathematical scientist in general. Suppose k particles X1, . . . , Xk diffuse in Euclidean n-space R n according to independent copies of an Ornstein-Uhlenbeck process. Thus X1, . . . , Xk obey the system (1.1) of stochastic differential equations