Piecewise-deterministic Markov process ($L_{t}f=a(t,\cdot )f^{\prime }+\alpha (K_{t}f-f)$)

A Markov process following a deterministic flow of a vector field between random jumps, with jump hazard function $\alpha (X_{t-})$ and destination transition kernel $K_t(X_{t-},dy)$. The integrated rate along the flowing path determines the next jump time. Retaining instead the holding times of a different piecewise-constant process need not yield a Markov process in the flowing state. Assume nonexplosion when describing paths at all finite times.

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.

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.

For a piecewise-deterministic Markov process, conditional on no jump before $u$, the state is $z(u)=\phi (s,x,u)$ and the survival probability is $S(u)=\exp [-{\int }_s^u\alpha (z(r))dr]$. The first jump has probability density function $\alpha (z(u))S(u)$; conditioning on its time and its destination gives $P_{s,t}f(x)=S(t)f(z(t))+{\int }_s^t\alpha (z(u))S(u)\int P_{u,t}f(y)K_u(z(u),dy)du$. This proves the identity for bounded Borel measurable functions under nonexplosion.