On adjoint additive processes

Kristian P. Evans

Niels Jacob

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 \(\delta_{\psi,t}\).

2010 AMS Mathematics Subject Classification: Primary 60J30, 60J35, 60E07, 60E10, 47D03, 47D06; Secondary XXXX.

Keywords and phrases: additive processes, L\'evy processes, adjoint densities, transition functions,
metric measure spaces.