 Forward equation of a piecewise-deterministic Markov process

If the destination transition kernel has probability density function $m_t(x,y)$, the formal adjoint operator of $L_{t}f=a{f}^{\prime }+\alpha (K_{t}f-f)$ is $(L_t^{\ast }p)(x)=-{\partial }_x(a(t,x)p(x))-\alpha (x)p(x)+\int \alpha (y)m_t(y,x)p(y)dy$. Multiply by a smooth compactly supported test function, use integration by parts on transport and Fubini's theorem on the jump gain to prove the formula when the integrals are finite. The general formulation is $\frac{d}{dt}\int fd{\nu }_t=\int L_{t}fd{\nu }_t$ for probability measures ${\nu }_t$. A deterministic initial value leaves positive no-jump atomic mass at finite times, so a full smooth density cannot be assumed for that initialization.