# Existence and uniqueness results for time-inhomogeneous time-change equations and Fokker--Planck equations

## TL;DR

This paper establishes the existence and uniqueness of solutions for certain time-inhomogeneous Fokker-Planck equations with degenerate coefficients, expanding the theoretical understanding of these equations under less restrictive conditions.

## Contribution

It provides new existence and uniqueness results for Fokker-Planck equations with degenerate, time-inhomogeneous coefficients without requiring global boundedness.

## Key findings

- Proved existence and uniqueness of solutions under new conditions.
- Developed a method using random time-changes and martingale problems.
- Extended the class of Fokker-Planck equations with known well-posedness.

## Abstract

We prove existence and uniqueness of solutions to Fokker--Planck equations associated to Markov operators multiplicatively perturbed by degenerate time-inhomogeneous coefficients. Precise conditions on the time-inhomogeneous coefficients are given. In particular, we do not necessarily require the coefficients to be neither globally bounded nor bounded away from zero. The approach is based on constructing random time-changes and studying related martingale problems for Markov processes with values in locally compact, complete and separable metric spaces.