Killed deterministic flow ($P_{s,t}^{\alpha }f(x)=f(\phi (s,x,t))e^{-{\int }_s^t\alpha (\phi (s,x,u))du}$)

The no-jump part of a piecewise-deterministic Markov process. The chain rule gives ${\partial }_t{P}_{s,t}^{\alpha }f=P_{s,t}^{\alpha }{L}_t^{\alpha }f$, where $L_t^{\alpha }f=a(t,\cdot )f^{\prime }-\alpha f$. The time subscript is essential for a time-dependent vector field.