= Forward equation of a piecewise-deterministic Markov process
{title2=$\partial_tp=-\partial_x(ap)-\alpha p+\int\alpha(y)m_t(y,x)p(y)dy$}
If the destination <transition kernel> has <probability density function> $m_t(x,y)$, the formal <adjoint operator> of $L_tf=af'+\alpha(K_tf-f)$ is $(L_t^*p)(x)=-\partial_x(a(t,x)p(x))-\alpha(x)p(x)+\int\alpha(y)m_t(y,x)p(y)dy$. 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 $\frac d{dt}\int f\,d\nu_t=\int L_tf\,d\nu_t$ 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.
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