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A large class of physically important nonlinear and nonhomogeneous evolutionproblems, characterized by advection-like and diffusion-like processes, can beusefully studied by a time-differential form of Kolmogorov's solution of thebackward-time Fokker-Planck equation. The differential solution embodies anintegral representation theorem by which any physical or mathematical entitysatisfying a generalized nonhomogeneous advection-diffusion equation can becalculated incrementally in time. The utility of the approach for tacklingnonlinear problems is illustrated via solution of the noise-free Burgers andrelated Kardar-Parisi-Zhang (KPZ) equations where it is shown that thedifferential Kolmogorov solution encompasses, and allows derivation of, theclassical 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.

Stochastic solution of nonlinear and nonhomogeneous evolution problems by a differential Kolmogorov equation