# Measure solutions to the conservative renewal equation

## TL;DR

This paper establishes the existence, uniqueness, and exponential convergence of measure solutions to the conservative renewal equation using a duality approach and Doeblin's argument.

## Contribution

It introduces a novel duality-based construction of measure solutions and applies Doeblin's argument to prove exponential relaxation to equilibrium.

## Key findings

- Existence and uniqueness of measure solutions proven.
- Solutions exhibit exponential relaxation to equilibrium.
- Duality approach effectively analyzes long-term behavior.

## Abstract

We prove the existence and uniqueness of measure solutions to the conservative renewal equation and analyze their long time behavior. The solutions are built by using a duality approach. This construction is well suited to apply the Doeblin's argument which ensures the exponential relaxation of the solutions to the equilibrium.