  The Brownian motion of a particle in a harmonic potential, which is
simultaneously exposed either to a linear shear flow or to a plane Poiseuille
flow is investigated. In the shear plane of both flows the probability
distribution of the particle becomes anisotropic and the dynamics is changed in
a characteristic manner compared to a trapped particle in a quiescent fluid.
The particle distribution takes either an elliptical or a parachute shape or a
superposition of both depending on the mean particle position in the shear
plane. Simultaneously, shear-induced cross-correlations between particle
fluctuations along orthogonal directions in the shear plane are found. They are
asymmetric in time. In Poiseuille flow thermal particle fluctuations
perpendicular to the flow direction in the shear plane induce a shift of the
particle's mean position away from the potential minimum. Two complementary
methods are suggested to measure shear-induced cross-correlations between
particle fluctuations along orthogonal directions.
