Factorized Duality, Stationary Product Measures and Generating Functions

Abstract

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.

Orthogonal Dualities of Markov Processes and Unitary Symmetries

“…However, orthogonality can be inferred by proving that the symmetry is unitary. A family of unitary symmetries will be found in Section 3 and, by specializing to some values of the parameters, we will recover orthogonal dualities in terms of discrete orthogonal polynomials previously found in [12,26]. This orthogonality task is also addressed in Section 4 where we show that biorthogonality can be achieved by construction.…”

“…These orthogonal dualities were identified as classical orthogonal polynomials in [12] by using the structural properties of those polynomials (recurrence relation and raising/lowering operators). In [26] the approach of generating functions was used instead, by which non-polynomial orthogonal dualities (provided by some other special functions, e.g., Bessel functions) were also found. Orthogonal duality functions can also be explained using representation theory: they can be understood as the intertwiner between two unitarily equivalent representations of a Lie algebra [13,17].…”

Quantitative Boltzmann–Gibbs Principles via Orthogonal Polynomial Duality

“…For more details on orthogonal duality and a proof of self-duality with respect to this function we refer to [7] and [11]. In those papers a more complete study is provided, which includes the case of other processes such as exclusion and inclusion, among others.…”

“…[11]) combined with precise estimates (of local limit type) of the n particle dynamics. Therefore, the results immediately apply in the context of the stationary symmetric exclusion process, and more generally for particle systems where these precise estimates (of local limit type) of the n particle dynamics can be obtained (e.g.…”

Duality in stochastic processes from the viewpoint of basis expansions

“…Recently, some mathematical discussions for the derivation of dual functions and dual processes have been done; in [10], recent developments have been reviewed. Although there are discussions focusing on duality functions [11], the derivations have been mainly performed by using mathematical properties of time-evolution operators (generators); the symmetry of the generators has been used to derive dual functions [12]. In [13,14], the discussion based on the second-quantization method (the Doi-Peliti formalism) for the time-evolution operators has also been given.…”