Basics and Applications of the Lie Algebraic Approach to Markov Duality

• Frank Redig (TU Delft)

Room: Centre de conférences Marilyn et James Simons Duality is a powerful tool in the study of large-scale stochastic systems (interacting particle systems). These interacting particle systems are used to model phenomena in non-equilibrium statistical physics, as well as in mathematical population genetics. Duality allows to relate complicated interacting systems to simpler systems, e.g. systems with only a few particles, or systems where non-equilibrium driving has been replaced by absorbing boundaries. In the last decade, a lot of work has been done to relate duality of Markov processes to symmetries of the underlying Markov generator, and from there one is naturally lead towards simple objects in Lie representation theory such as the Casimir, coproduct, tensor products, which become tools to constructively find processes with useful dualities. For symmetric processes (detailed balance setting) these Lie algebras are classical ones (such as SU(2), SU(1,1), SU(n), SU(n,n)), whereas for the corresponding asymmetric processes, their quantum deformations are used. In this seminar, starting with simple examples, we show how the Lie-algebraic approach allows to unify several processes, and provides a systematic method to produce processes which have some degree of exact solvability. We start with the classical processes such as independent random walkers and the symmetric inclusion process. We will also discuss recent developments in the representation of non-equilibrium states as mixtures of product measures, and how this can be used to study fluctuations of the density profile for such systems.

From Markov Duality to Discrete Macroscopic Fluctuation Theory

• Cristian Giardinà (Università di Modena e Reggio Emilia)

Room: Centre de conférences Marilyn et James Simons Building on the Lie-algebraic approach to duality, I will revisit the harmonic process and show that it arises as the Markov process associated with the integrable XXX_s Hamiltonian with non-compact spins. Its large-spin limit yields a lattice formulation of macroscopic fluctuation theory whose large-deviation Hamiltonian is the classical Ishimori chain. Exploiting this integrable structure, we solve the current large-deviation problem on the infinite line with step initial data (joint work with K. Mallick, H. Suda, and T. Sasamoto).

Applications of Duality: Forward and Backward Processes in Mathematical Population Genetics

• Noemi Kurt (Goethe-Universität Frankfurt)

Room: Centre de conférences Marilyn et James Simons In this talk we will put the main focus on applications of the concept of duality to quite concrete stochastic models. In particular, we will shift focus from the abstract duality framework on the level of generators to consider stochastic processes in a pathwise set up. We will discuss how dual processes may often be interpreted in a strong pathwise sense as forward- and backward pictures of the same underlying stochastic models. This is particularly evident in many classical models of population genetics, such as the Wright-Fisher model, where one process describes the forward evolution of an allele frequency, and the dual process its genealogy. Thanks to this connection, duality may be applied to easily calculate quantities of interest such as fixation probabilities, as we will demonstrate. We will discuss several such applications in classical population genetics and in population models with dormancy, as well as in interacting particle systems, and discuss connections between strong pathwise dualities and generator dualities.