We study three classes of continuous time Markov processes (inclusion process, exclusion process, independent walkers) and a family of interacting diffusions (Brownian energy process). For each model we define a boundary driven process which is obtained by placing the system in contact with proper reservoirs, working at different particle densities or different temperatures. We show that all the models are exactly solvable by duality, using a dual process with absorbing boundaries. The solution does also apply to the so-called thermalization limit in which particles or energy is instantaneously redistributed among sites.The results shows that duality is a versatile tool for analyzing stochastic models of transport, while the analysis in the literature has been so far limited to particular instances. Long-range correlations naturally emerge as a result of the interaction of dual particles at the microscopic level and the explicit computations of covariances match, in the scaling limit, the predictions of the macroscopic fluctuation theory.

We study a new process, which we call ASEP(q, j), where particles move asymmetrically on a one-dimensional integer lattice with a bias determined by q ∈ (0, 1) and where at most 2 j ∈ N particles per site are allowed. The process is constructed from a (2 j + 1)-dimensional representation of a quantum Hamiltonian with U q (sl 2 ) invariance by applying a suitable ground-state transformation. After showing basic properties of the process ASEP(q, j), we prove self-duality with several selfduality functions constructed from the symmetries of the quantum Hamiltonian. By making use of the self-duality property we compute the first q-exponential moment of the current for step initial conditions (both a shock or a rarefaction fan) as well as when the process is started from a homogeneous product measure. Mathematics Subject Classification

We apply our general method of duality, introduced in [15], to models\ud of population dynamics. The classical dualities between forward\ud and ancestral processes can be viewed as a change of representation\ud in the classical creation and annihilation operators, both for diffusions\ud dual to coalescents of Kingman’s type, as well as for models with finite\ud population size.\ud Next, using SU(1, 1) raising and lowering operators, we find new\ud dualities between the Wright-Fisher diffusion with d types and the\ud Moran model, both in presence and absence of mutations. These new\ud dualities relates two forward evolutions. From our general scheme we\ud also identify self-duality of the Moran model

By using the algebraic construction outlined in Carinci et al. (arXiv:1407.3367, 2014, we introduce several Markov processes related to the U q (su(1, 1)) quantum Lie algebra. These processes serve as asymmetric transport models and their algebraic structure easily allows to deduce duality properties of the systems. The results include: (a) the asymmetric version of the Inclusion Process, which is self-dual; (b) the diffusion limit of this process, which is a natural asymmetric analogue of the and which turns out to have the Symmetric Inclusion Process as a dual process; (c) the asymmetric analogue of the KMP Process, which also turns out to have a symmetric dual process. We give applications of the various duality relations by computing exponential moments of the current.

We study a system of particles in the interval [0, −1 ] ∩ Z, −1 a positive integer. The particles move as symmetric independent random walks (with reflections at the endpoints); simultaneously new particles are injected at site 0 at rate j ( j > 0) and removed at same rate from the rightmost occupied site. The removal mechanism is, therefore, of topological rather than metric nature. The determination of the rightmost occupied site requires a knowledge of the entire configuration and prevents from using correlation functions techniques. We prove using stochastic inequalities that the system has a hydrodynamic limit, namely that under suitable assumptions on the initial configurations, the law of the density fields φ( x)ξ −2 t (x) (φ a test function, ξ t (x) the number of particles at site x at time t) concentrates in the limit → 0 on the deterministic value φρ t , ρ t interpreted as the limit density at time t. We characterize the limit ρ t as a weak solution in terms of barriers of a limit-free boundary problem.

We study self-duality for interacting particle systems, where the particles move as continuous time random walkers having either exclusion interaction or inclusion interaction. We show that orthogonal self-dualities arise from unitary symmetries of the Markov generator. For these symmetries we provide two equivalent expressions that are related by the Baker-Campbell-Hausdorff formula. The first expression is the exponential of an anti Hermitian operator and thus is unitary by inspection; the second expression is factorized into three terms and is proved to be unitary by using generating functions. The factorized form is also obtained by using an independent approach based on scalar products, which is a new method of independent interest that we introduce to derive (bi)orthogonal duality functions from non-orthogonal duality functions.

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We study the so-called nonconventional averages in the context of lattice spin systems, or equivalently random colourings of the integers. For i.i.d. colourings, we prove a large deviation principle for the number of monochromatic arithmetic progressions of size two in the box [1, N ] ∩ N, as N → ∞, with an explicit rate function related to the one-dimensional Ising model. For more general colourings, we prove some bounds for the number of monochromatic arithmetic progressions of arbitrary size, as well as for the maximal progression inside the box [1, N ] ∩ N. Finally, we relate nonconventional sums along arithmetic progressions of size greater than two to statistical mechanics models in dimension larger than one.