On Adjoint Additive Processes

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 $\delta_{\psi,t}$.