Isotropic Brownian flows on the sphere.

We study isotropic Brownian flows on homogeneous spaces and particularly on the sphere Sd-l. An isotropic Brownian flow is directed by xi an isotropic Gaussian vector field and therefore is characterized by a covariance matri x. Using the irreducible representations of SO(d), we calculate this covarianc e matrix. Given this covariance matrix, we compute the Lyapounov exponents of the flow, which describe its asymptotic behavior, In particular, we see that for d less than or equal to 5 a gradient flow is always stable. (C) El sevier, Paris.