Advances in Stochastic Duality for Interacting Particle Systems: from many to few

Interacting particle systems (IPS) is a subfield of probability theory that provided a fruitful framework in which several questions of physical interests have been answered with mathematical rigor. An interacting particle system is a stochastic system consisting of a very large number of particles interacting with each other. The class of IPS considered in this manuscript is the one of systems satisfying stochastic duality. Stochastic duality is a useful tool in probability theory which allows to study a Markov process (the one that interests you) via another Markov process, called dual process, which is hopefully easier to be studied. The connection between the two processes is established via a function, the so-called duality function, which takes as input configurations of both processes. In the context of IPS, one of the typical simplifications provided by stochastic duality is that a system with an infinite number of particles can be studied via a finite number of particles (the simplification from many to few).
In this thesis, we extend the theory and the applications of stochastic duality in the following two contexts:
i) evolution of particles in space inhomogeneous settings and more precisely, processes in random environment
and processes in a multi-layer system;
ii) evolutions of particles in the continuum.