On weak Brownian motions of arbitrary order

We show the existence, for any k is an element of N, of processes which hav e the same k-marginals as Brownian motion, although they are not Brownian m otions. For k = 4, this proves a conjecture of Stoyanov. The law (P) over t ilde of such a "weak Brownian motion of order k" can be constructed to be e quivalent to Wiener measure P on C[0, 1]. On the other hand, there are weak Brownian motions of arbitrary order whose law is singular to Wiener measur e. We also show that, for any epsilon > 0, there are weak Brownian motions whose law coincides with Wiener measure outside of any interval of length e psilon. (C) 2000 Editions scientifiques et medicales Elsevier SAS.