Title: On spatial vs temporal descriptions of stochastic processes
Abstract: Markov chains with discrete state space are often defined via their transition matrix as temporal processes. Under fairly general assumptions, Markov chains can equivalently be described as Markov fields, i.e. spatial processes. An immediate follow-up question is if a corresponding equivalence between temporal and spatial processes holds for more general classes of stochastic processes. In this talk, I will first review recent developments showing that, for the natural generalization of Markovian to instead require continuous dependency on the past/boundary, this is in general not the case. Then I will focus on two concrete examples: the Ising model and the contact process (also known as the SIS model). These models contain natural examples of (hidden) stochastic processes which do not posses a “well-behaved” spatial description. I will present recent work on their temporal descriptions, partly based on joint work with Diana Conache (TUM).
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