# On the fundamental solution and a variational formulation of a degenerate diffusion of Kolmogorov type

## TL;DR

This paper constructs the fundamental solution for a degenerate Kolmogorov diffusion and introduces a variational scheme for its adjoint, leveraging optimal transport and mean squared derivative costs, with proven convergence.

## Contribution

It provides a novel fundamental solution and a variational scheme for the adjoint of a degenerate Kolmogorov diffusion, extending previous methods.

## Key findings

- Successfully constructed the fundamental solution.
- Developed a convergent variational scheme.
- Extended results to more general degenerate diffusions.

## Abstract

In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. We establish the convergence of the scheme to the solution of the adjoint equation generalizing previously known results for the Fokker-Planck equation and the Kramers equation.