Path sums for Brownian motion and quantum mechanics

A treatment is given of classical Brownian motion in phase space based on path summation. It treats efficiently the usual exactly solvable cases when the external force is linear in momentum or position. The method might be useful for generating approximations for more complicated external forces. A path sum formalism is given to generate the Wigner propagator in the Wigner-Weyl phase space formulation of quantum mechanics. The short-time Brownian and Wigner propagators bear a generic similarity.