Brownian motions on shape and size-and-shape spaces

The diffusions on the shape and size-and-shape spaces induced by brownian motions on the pre-size-and-shape spaces have been investigated in several papers (cf. [1], [6], [7]). We here address the dual problem: the character of the diffusions on the pre-shape and pre-size-and-shape spaces which induce brownian motions on the shape and size-and-shape spaces. In particular we show that the shape and size-and-shape spaces for k labelled points in R(m) are stochastically complete if k > m and obtain the heat kernels of certain diffusions which induce brownian motions on the size-and-shape spaces.