Existence theorems for nonlinear stationary Kolmogorov equations with partially degenerate diffusion matrices

TL;DR

This paper proves the existence of solutions for nonlinear stationary Kolmogorov equations with degenerate diffusion matrices, introducing a new approach using Lyapunov functions and solution projections.

Contribution

It presents a novel method for establishing solutions in degenerate cases, expanding the theoretical understanding of such equations.

Findings

Existence of solutions is established for the equations.

A new approach based on integral conditions and Lyapunov functions is proposed.

Examples illustrate the applicability of the theoretical results.

Abstract

We study nonlinear stationary Kolmogorov equations with degenerate diffusion matrices and discontinuous coefficients. The existence of a solution is proved. We propose a new approach based on an integral condition with Lyapunov functions and a regularity of projections of solutions in the partially degenerate case. Examples are given to illustrate the results.