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Let $N(\tau)$ be a renewal process for independent holding times $\{X_i\}_{k\ge 0}$ ,where $\{X_k\}_{k\ge 1}$ are identically distributed with density$p(x)$. If the associated residual time $R(\tau)$ has a density $u(x,\tau)$,its Kolmogorov forward equation is given by \begin{equation*} \partial_\tauu(x,\tau)-\partial_x u(x,\tau) = p(x)u(0,\tau), \quad x,\tau \in [0, \infty),\end{equation*} with an initial holding time density $u(x,0)=u_0(x)$. We derivea measure-valued solution formula for the density of residual times after anexpected number of renewals occur. Solutions under this time scale are thenshown to evolve continuously in the space of measures with the weak topologyfor a wide variety of holding times.

Renewal-scaled solutions of the Kolmogorov forward equation for residual times