Recently, a new trajectory method for advancing solutions of evolutionary partial differential equations was proposed: the derivative propagation method (DPM) evolves the solution and its spatial derivatives concurrently and eliminates the need for function fitting, finite-differences (FDs), etc. This trajectory method and a stationary lattice FD algorithm are applied in phase space to solve the classical Klein-Kramers and quantum modified Caldeira-Leggett equations for several examples: a double-well oscillator in contact with a thermal bath and the decay of a metastable state. For the latter potential, the trajectory and fixed-grid solutions are compared and any discrepancies are noted. (C) 2004 Elsevier B.V. All rights reserved.