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

Manh Hong Duong; Hoang Minh Tran

doi:10.3934/dcds.2018146

Article Contents

2018, Volume 38, Issue 7: 3407-3438. Doi: 10.3934/dcds.2018146

This issue Previous Article Navier-Stokes-Oseen flows in the exterior of a rotating and translating obstacle Next Article Non-floquet invariant tori in reversible systems

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

Manh Hong Duong^1, , and
Hoang Minh Tran^2,

1.
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
2.
Data Analytics Department, Esmart Systems, Håkon Melbergs vei 16, 1783 Halden, Norway

* Corresponding author: Manh Hong Duong
* Corresponding author: Manh Hong Duong

Received: July 2017

Revised: November 2017

Published: July 2018

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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.

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Mathematics Subject Classification: Primary: 35K15, 35A15; Secondary: 35Q99, 37J99, 60J60.

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