The steady transport of Brownian particles convected by a periodic flow field is studied by following the motion of a randomly chosen tagged particle in an otherwise uniform solute concentration field. A nonlocal, Fickian constitutive relation is derived, in which the steady mass flux of Brownian particles equals a convolution integral of the concentration gradient times a (tensorial) diffusion functionDL(R). In turn, the diffusion function is uniquely determined via thenth diffusivities, which are determined analytically in terms of thenth cumulants of the probability distribution by exploiting the translational symmetry of the velocity field. The Lagrangian, long‐time self‐diffusion functionDL(R) is shown to be equal to the symmetric part of the Eulerian, gradient diffusion functionDE(R). Since the latter characterizes the dissipative steady‐state mass transport, whileDL(R) describes the fluctuations of the concentration field about its uniform equilibrium value, the equality betweenDE(R) andDL(R) can be seen as an aspect of the fluctuation–dissipation theorem. Finally, the present results are applied to study the transport of solute particles immersed in a fluid flowing in rectilinear pipes and through periodic fixed beds of spheres at low Pe´clet number. In the first case, the first sixnth diffusivities are determined; in the second, the first two diffusivities are calculated, showing that the enhancement to the second diffusivity due to convection is eight times larger in the direction parallel to the fluid flow than in the transversal direction. ©1995 American Institute of Physics.