Negative "probabilities" (in the theory of duality of Markov chains) and set-valued dual chains

We will present preliminary results on i) negative "probabilities" (focusing on duality in Markov chains) and ii) set-valued dual Markov chains. i) Paul Dirac and Richard Feynman have argued for the validity and utility of negative "probabilities." While these have been useful tools for solving complex problems, they have not been associated with real processes. We propose a method for "realizing" processes that represent transition probabilities with "negative entries." This method emerged from work on dualities in Markov chains, aiding in modeling and simulating processes where traditional methods fall short, especially in systems lacking monotonicity. However, its applications are potentially much broader. ii) To address problems for a given Markov chain (e.g., winning probability, rate of convergence to stationarity), we can compute a dual chain (strong stationary dual or Siegmund dual) to solve potentially easier problem (e.g., stationary distribution or absorption time). For chains lacking specific types of monotonicity, instead of resorting to negative "probabilities," we demonstrate a construction of a set-valued dual chain that always exists. This talk is a preliminary report on joint work with Bartłomiej Błaszczyszyn (INRIA, Paris). It will mainly present the ideas and methods, applied to 2 or 3 state chains.