Equation-free approaches have been proposed in recent years for the computational study of multiscale phenomena in engineering problems where evolution equations for the coarse-grained, system-level behavior are not explicitly available. In this paper, we study the dynamics of a diffusive particle system in a laminar shear flow, described by a two-dimensional Brownian motion; in particular, we perform coarse projective integration and demonstrate the particle-based computation of coarse self-similar and asymptotically self-similar solutions for this problem. We use marginal and conditional inverse cumulative distribution functions ( ICDFs) as the macroscopic observables of the evolving particle distribution.