Side Boundary potentials for a Kolmogorov-type PDE

TL;DR

This paper introduces a novel method for solving a Kolmogorov-type hypoelliptic PDE with boundary conditions by constructing an approximate boundary potential and formulating a boundary-domain Volterra equation, revealing boundary effects and periodic behaviors.

Contribution

It develops a new approach using boundary potentials and Volterra equations to solve hypoelliptic PDEs with boundary conditions, including polynomial corrections for improved accuracy.

Findings

Constructed an approximate boundary potential capturing boundary effects.

Formulated a boundary-domain Volterra integral equation for the PDE.

Revealed periodic behavior in solution bounds due to boundary effects.

Abstract

We solve a Kolmogorov-type hypoelliptic parabolic partial differential equation with a "side" boundary condition (in the direction of the weak H\"ormander condition). We construct an approximate boundary potential which captures the effect of the boundary condition. Integrals against this approximate boundary potential have a novel discontinuity at the boundary. We introduce some polynomial corrections to this approximate boundary potential and then construct a boundary-domain Volterra equation to solve the original partial differential equation. This Volterra integral equation is iteratively solved, and the bounds contain a periodic behavior resulting from the boundary effects.