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

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.

💡 Research Summary

The paper introduces a novel stochastic framework for solving a broad class of nonlinear and non‑homogeneous evolution equations that are governed by advection‑like and diffusion‑like processes. The authors start from Kolmogorov’s backward‑time Fokker‑Planck equation and, instead of using its integral form directly, they differentiate it with respect to time. This yields a “differential Kolmogorov solution” that provides an incremental representation: for a small time interval Δt the solution u(x,t+Δt) can be expressed as the expectation of u evaluated along stochastic trajectories driven by the underlying drift a(x,t) and diffusion D(x,t), plus an integral of any source term f(x,t) over the same interval. In compact notation,

 u(x,t+Δt)=E