**Short Talks**

#### Chiara Franceschini

Two duality relations for Markov processes with an open boundaryIn this talk I will show how the same algebraic approach, which relies on the \(SU(1,1)\) Lie algebra, can be used to construct two duality results. One is well-known: the two processes involved are the symmetric inclusion process and a Markov diffusion called Brownian Energy process. The other one is a new result which involves a particle system of zero-range type, called harmonic process, and a redistribution model similar to the Kipnis-Marchioro-Presutti model. Despite the similarity, it turns out that the second relation involves integrable models and thus duality can be pushed further. As a consequence, all moments in the stationary non-equilibrium state can be explicitly computed.

#### Carla Crucianelli

Graphon particle systems and its large population limitWe consider a general interacting particle system with interactions on a random graph and study the large population limit of this system. When the sequence of underlying graphs converges to a graphon, we show convergence of the interacting particle system to a so called graphon stochastic differential equation. This is a system of uncountable many SDEs of McKean-Vlasov type driven by a continuum of Brownian motions. We make sense of this equation in a way that retains joint measurability and essentially pairwise independence of the driving Brownian motions of the system by using the framework of Fubini extensions. The convergence result is general enough to cover nonlinear interaction, as well as various examples of sparse graphs (joint work with L. Tangpi).

#### Daniel Yukimura

Quantitative results on sampling from quasi-stationary distributionsWe study the problem of approximating the quasi-stationary distribution (QSD) of Markov processes using simulation methods. I will present both a lower bound and an upper bound for the rate of convergence of these methods to the QSD. We show that the rates of convergence depend on intrinsic properties of the process, like a notion of mixing time, and a uniform notion of how fast the process gets killed. Finally, we apply these to get quantitative estimates for a large class of killed processes in \(\mathbb{Z}^d\) and obtain bounds on a ''natural'' time scale.

#### Angeliki Koutsimpela

Tagged particles and size-biased dynamics in mean-field interacting particle systemsWe establish a connection between tagged particles and size-biased empirical processes in interacting particle systems, in analogy to classical results on the propagation of chaos. In a mean-field scaling limit, the evolution of the occupation number on the tagged particle site converges to a time-inhomogeneous Markov process with non-linear master equation given by the law of large numbers of size-biased empirical measures.

#### María Cecilia De Vita

Synchronization in Random Geometric GraphsThe Kuramoto model is a system of ordinary differential equations that describes the behavior of coupled oscillators. In this talk, we consider the case where the coupling is given by a random geometric graph on the unit circle. We ask whether it is possible to guarantee global synchronization (for almost any initial condition, the system converges to a state where all phases coincide) or if there exist other stable equilibria. To address this question, we work with the energy function of the Kuramoto model on these graphs and prove the existence of at least one local minimum for each winding number \(q\) (with high probability). There is a correspondence between these states and the explicit 'twisted states' found in cycle graphs, although in this case without an explicit formula.