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 $\delta_{ \psi,t}$.
Categories: math.FA