# On Adjoint Additive Processes

## TL;DR

This paper explores the construction of an adjoint additive process from a given additive process and analyzes how its transition densities are controlled by related metrics, revealing a duality in their properties.

## Contribution

It introduces a method to construct an adjoint additive process and demonstrates how its transition densities relate to the original process through metric control.

## Key findings

- Construction of an adjoint additive process from a given process
- Transition densities of the adjoint process are controlled by inverted metrics
- Establishes a duality relationship between original and adjoint processes

## Abstract

Starting with an additive process $(Y_t)_{t\geq0}$, it is in certain cases possible to construct an adjoint process $(X_t)_{t\geq0}$ which is itself additive. Moreover, assuming that the transition densities of $(Y_t)_{t\geq0}$ are controlled by a natural pair of metrics $\mathrm{d}_{\psi,t}$ and $\delta_{\psi,t}$, we can prove that the transition densities of $(X_t)_{t\geq0}$ are controlled by the metrics $\delta_{\psi,1/t}$ replacing $\mathrm{d}_{\psi,t}$ and $\mathrm{d}_{\psi,1/t}$ replacing $\delta_{\psi,t}$.