Markovianity in space and time

Markov chains in time, such as simple random walks, are at the heart of probability. In space, due to the absence of an obvious definition of past and future, a range of definitions of Markovianity have been proposed. In this paper, after a brief review, we introduce a new concept of Markovianity that aims to combine spatial and temporal conditional independence. 1. From Markov chain to Markov point process, and beyond This paper is devoted to the fundamental concept of Markovianity. Although its precise definition depends on the context, common ingredients are conditional in- dependence and factorisation formulae that allow to break up complex, or high dimensional, probabilities into manageable, lower dimensional components. Thus, computations can be greatly simplified, sometimes to the point that a detailed probabilistic analysis is possible. If that cannot be done, feasible, efficient simula- tion algorithms that exploit the relatively simple building blocks may usually be designed instead.