Global in Time Madelung Transformation for Kolmogorov-Feller Pseudodifferential Equations

TL;DR

This paper develops a global in time solution framework for the transport equation related to Kolmogorov-Feller equations, incorporating diffusion, potential, and jump processes, using a generalized delta-shock approach.

Contribution

It introduces a novel method for constructing global solutions to the transport equation with discontinuous velocity fields in the context of Kolmogorov-Feller equations.

Findings

Constructed global in time solutions for the transport equation.

Extended Madelung's idea to systems with jumps and diffusion.

Discussed asymptotic solution construction via Maslov tunnel asymptotics.

Abstract

Using an idea going back to Madelung we construct global in time solutions to the transport equation corresponding to the asymptotic solution of the Kolmogorov-Feller equation describing a system with diffusion, potential and jump terms. To do that we use the construction of a generalized delta -shock solution of the continuity equation for a discontinuous velocity field. We also discuss corresponding problem of asymptotic solution construction (Maslov tunnel asymptotics).