We find all self-duality functions of the form for a class of interacting particle systems. We call these duality functions of simple factorized form. The functions we recover are self-duality functions for interacting particle systems such as zero-range processes, symmetric inclusion and exclusion processes, as well as duality and self-duality functions for their continuous counterparts. The approach is based on, firstly, a general relation between factorized duality functions and stationary product measures and, secondly, an intertwining relation provided by generating functions. For the interacting particle systems, these self-duality and duality functions turn out to be generalizations of those previously obtained in Giardinà et al. (J Stat Phys 135:25–55, 2009 ) and, more recently, in Franceschini and Giardinà (Preprint, arXiv:1701.09115 , 2016 ) . Thus, we discover that only these two families of dualities cover all possible cases. Moreover, the same method discloses all simple factorized self-duality functions for interacting diffusion systems such as the Brownian energy process, where both the process and its dual are in continuous variables.

We consider the symmetric simple exclusion process in Z d with quenched bounded dynamic random conductances and prove its hydrodynamic limit in path space. The main tool is the connection, due to the self-duality of the process, between the invariance principle for single particles starting from all points and the macroscopic behavior of the density field. While the hydrodynamic limit at fixed macroscopic times is obtained via a generalization to the time-inhomogeneous context of the strategy introduced in [41], in order to prove tightness for the sequence of empirical density fields we develop a new criterion based on the notion of uniform conditional stochastic continuity, following [50]. In conclusion, we show that uniform elliptic dynamic conductances provide an example of environments in which the so-called arbitrary starting point invariance principle may be derived from the invariance principle of a single particle starting from the origin. Therefore, our hydrodynamics result applies to the examples of quenched environments considered in, e.g., [1], [3], [6] in combination with the hypothesis of uniform ellipticity.

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We start from the observation that, anytime two Markov generators share an eigenvalue, the function constructed from the product of the two eigenfunctions associated to this common eigenvalue is a duality function. We push further this observation and provide a full characterization of duality relations in terms of spectral decompositions of the generators for finite state space Markov processes. Moreover, we study and revisit some well-known instances of duality, such as Siegmund duality, and extract spectral information from it. Next, we use the same formalism to construct all duality functions for some solvable examples, i.e., processes for which the eigenfunctions of the generator are explicitly known.

We study the "Immediate Exchange Model", a wealth distribution model introduced in Heinsalu and Patriarca (Eur Phys J B 87:170, 2014). We prove that the model has a discrete dual, where the duality functions are natural polynomials associated to the Gamma distribution with shape parameter 2 and are exactly those connecting the Brownian Energy Process As a consequence, we recover invariance of products of Gamma distributions with shape parameter 2, and obtain ergodicity results. Next we show similar properties for a more general model, where the exchange fraction is Beta(s, t) distributed, and product measures with Gamma(s + t) marginals are invariant. We also show that the discrete dual model itself is self-dual and has the original continuous model as its scaling limit. We show that the selfduality is linked with an underlying SU (1, 1) symmetry, reminiscent of the one found before for the Symmetric Inclusion Process and related processes.