Existence of a unique strong solution to the DMPK equation

TL;DR

This paper proves the well-posedness of the DMPK equation, a stochastic differential equation modeling electron transmission in disordered materials, by controlling singularities and initial degeneracies.

Contribution

It establishes the existence and uniqueness of strong solutions to the DMPK equation, addressing challenges from singular repulsion and degenerate initial conditions.

Findings

DMPK equation is well-posed with controlled singularities.

Established strong solution existence and uniqueness.

Provided a method to handle degenerate initial conditions.

Abstract

For the transmission of electrons in a weakly disordered strip of material Dorokhov, Mello, Pereyra and Kumar (DMPK) proposed a diffusion process for the transfer matrices. The correspoding transmission eigenvalues satisfy the DMPK stochastic differential equations, like Dyson Brownian motion in the context of GOE/GUE random matrices. We control the singular repulsion terms of this SDE with a stopping-time argument, and its degenerate initial condition by an approximation procedure, and thereby establish the DMPK equation to be well posed.