On stochastic areas and averages of planar Brownian motion

Duplantier (1989) has computed the characteristic function of the stochastic area of planar Brownian motion taken with respect to its centre of gravity. In this paper it is shown that this result can be deduced from Levy's stochastic area formula. Moreover, Duplantier's stochastic area is shown to have the same law as the stochastic area of the restriction to the time interval (0, 1) of the Brownian ring defined on the time interval (0, 4/3). The same method also applies to the stochastic area formulae, obtained by Biane and Yor (1986), which are relative to certain planar processes associated with the orthogonal decomposition of Brownian motion along the L 2 (0, 1) basis of Legendre polynomials.