Patient Reaper is a notable Ascendancy passive skill for the Trickster that grants increased damage over time. It also grants an increase to life, mana and energy shield recovery if the character has killed an enemy affected by one of his damage over time effects recently (within the past four seconds).
Damage over time effects such as poison or ignite are applied after the initial hit is applied, since they depend on the size of the hit. Therefore, hits that kill the enemy instantly will not trigger Patient Reaper.
The increase in energy shield recharge rate will show on the character sheet, however passive regeneration (and, by extension, energy shield regeneration via Zealot's Oath) will not. The interaction has been confirmed by GGG and through independent testing.