Explicit Structure of the Fokker-Planck Equation with potential

TL;DR

This paper analyzes the detailed pointwise behavior of solutions to the Fokker-Planck equation with flat confinement, revealing how the potential influences large-time and asymptotic behaviors in the space-time plane.

Contribution

It provides a detailed description of the solution's structure, including large-time behavior, initial layer, and asymptotics, emphasizing the role of the potential function.

Findings

Explicit description of the solution's pointwise behavior

Characterization of large-time and asymptotic behaviors

Dependence of solution structure on the potential function

Abstract

We study the pointwise (in the space and time variables) behavior of the Fokker-Planck Equation with flat confinement. The solution has very clear description in the $xt-$plane, including large time behavior, initial layer and asymptotic behavior. Moreover, the structure of the solution highly depends on the potential function.