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

TL;DR

This paper introduces a stochastic, differential Kolmogorov approach to solving nonlinear and nonhomogeneous evolution equations, enabling incremental solutions and encompassing classical transformations like Cole-Hopf and KPZ.

Contribution

It presents a novel differential solution framework for complex evolution problems, unifying classical methods and extending applicability to nonlinear and nonhomogeneous cases.

Findings

Successfully applied to Burgers and KPZ equations

Derived the Feynman-Kac formula for nonhomogeneous problems

Unified classical transformations within the differential Kolmogorov framework

Abstract

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.