On $ h $-transforms of one-dimensional diffusions stopped upon hitting zero

Abstract

For a one-dimensional diffusion on an interval for which 0 is the regular-reflecting left boundary, three kinds of conditionings to avoid zero are studied. The limit processes are $ h $-transforms of the process stopped upon hitting zero, where $ h $'s are the ground state, the scale function, and the renormalized zero-resolvent. Several properties of the $ h $-transforms are investigated.