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The representation problem of finite-dimensional Markov matrices in Markov semigroups is revisited, with emphasis on concrete criteria for matrix subclasses of theoretical or practical relevance, such as equal-input, circulant, symmetric or doubly stochastic matrices. Here, we pay special attention to various algebraic properties of the embedding problem, and discuss the connection with the centraliser of a Markov matrix.
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
Moran or Wright-Fisher processes are probably the most well known model to study the evolution of a population under various effects. Our object of study will be the Simpson index which measures the level of diversity of the population, one of the ke
We reconsider the deterministic haploid mutation-selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (r
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 sampling and enumeration problems for Markov equivalence classes. We create and analyze a Markov chain for uniform random sampling on the DAGs inside a Markov equivalence class. Though the worst case is exponentially slow mixing, we find