Killed deterministic flow
= Killed deterministic flow
{title2=$P^\alpha_{s,t}f(x)=f(\varphi(s,x,t))e^{-\int_s^t\alpha(\varphi(s,x,u))du}$}
The no-jump part of a <piecewise-deterministic Markov process>. The <chain rule> gives $\partial_tP^\alpha_{s,t}f=P^\alpha_{s,t}L_t^\alpha f$, where $L_t^\alpha f=a(t,\cdot)f'-\alpha f$. The time subscript is essential for a time-dependent <vector field>.