  A large class of physically important nonlinear and nonhomogeneous evolution
problems, characterized by advection-like and diffusion-like processes, can be
usefully studied by a time-differential form of Kolmogorov's solution of the
backward-time Fokker-Planck equation. The differential solution embodies an
integral representation theorem by which any physical or mathematical entity
satisfying a generalized nonhomogeneous advection-diffusion equation can be
calculated incrementally in time. The utility of the approach for tackling
nonlinear problems is illustrated via solution of the noise-free Burgers and
related Kardar-Parisi-Zhang (KPZ) equations where it is shown that the
differential Kolmogorov solution encompasses, and allows derivation of, the
classical Cole-Hopf and KPZ transformations and solutions. A second example,
illustrating application of this approach to nonhomogeneous evolution problems,
derives the Feynman-Kac formula appropriate to a Schrodinger-like equation.
