  We reformulate a general class of classical bead-spring-chain models for
dilute polymeric fluids, with Hookean spring potentials, as McKean-Vlasov
diffusion. This results in a coupled system of partial differential equations
involving the unsteady incompressible linearized Navier-Stokes equations,
referred to as the Oseen system, for the velocity and the pressure of the
fluid, with a source term which is a nonlinear function of the probability
density function, and a second-order degenerate parabolic Fokker-Planck
equation, whose transport terms depend on the velocity field, for the
probability density function. We show that this coupled Oseen-Fokker-Planck
system has a large-data global weak solution. We then perform a rigorous
passage to the limit as the masses of the beads in the bead-spring-chain
converge to zero, which is shown in particular to result in equilibration in
momentum space. The limiting problem is then used to perform a rigorous
derivation of the Hookean bead-spring-chain model for dilute polymeric fluids,
which has the interesting feature that, if the flow domain is bounded, then so
is the associated configuration space domain and the associated Kramers stress
tensor is defined by integration over this bounded configuration domain.
