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Suppose that $(X_t)_{t ge 0}$ is a one-dimensional Brownian motion with negative drift $-mu$. It is possible to make sense of conditioning this process to be in the state $0$ at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to $0$, then the limit of the killed Markov process evolves like $X$ conditioned to hit $0$, after which time it behaves as $X$ killed at the last time $X$ visits $0$. Equivalently, the limit process has the dynamics of the killed bang--bang Brownian motion that evolves like Brownian motion with positive drift $+mu$ when it is negative, like Brownian motion with negative drift $-mu$ when it is positive, and is killed according to the local time spent at $0$. An extension of this result holds in great generality for Borel right processes conditioned to be in some state $a$ at an exponential random time, at which time they are killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the bang--bang construction for general Markov processes. As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the $h$-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.
Semi-Markov processes are a generalization of Markov processes since the exponential distribution of time intervals is replaced with an arbitrary distribution. This paper provides an integro-differential form of the Kolmogorovs backward equations for
The time it takes the fastest searcher out of $Ngg1$ searchers to find a target determines the timescale of many physical, chemical, and biological processes. This time is called an extreme first passage time (FPT) and is typically much faster than t
We consider a general piecewise deterministic Markov process (PDMP) $X={X_t}_{tgeqslant 0}$ with measure-valued generator $mathcal{A}$, for which the conditional distribution function of the inter-occurrence time is not necessarily absolutely continu
This paper describes the structure of solutions to Kolmogorovs equations for nonhomogeneous jump Markov processes and applications of these results to control of jump stochastic systems. These equations were studied by Feller (1940), who clarified in
We construct a family of genealogy-valued Markov processes that are induced by a continuous-time Markov population process. We derive exact expressions for the likelihood of a given genealogy conditional on the history of the underlying population pr