### 21/11/2017, 15:00 — 16:00 — Room P3.10, Mathematics Building

Leonardo de Carlo, *CAMGSD, Instituto Superior Técnico*

### Geometric and combinatoric structures in stationary Markov chains

We study the combinatoric and geometric structure of stationary non-reversible Markov chains defined on graphs, in particular in applications we focus on non-equilibrium states for interacting particle systems from a microscopic viewpoint. We discuss two different discrete geometric constructions. We apply both of them to determine non reversible transition rates corresponding to a fixed invariant measure. The first one uses the equivalence of this problem with the construction of divergence free flows on the transition graph. The second construction is a functional discrete Hodge decomposition for translational covariant discrete vector fields. This decomposition is unique and constructive. The stationary condition can be interpreted as an orthogonality condition with respect to an harmonic discrete vector field. This decomposition applied to the instantaneous current of any interacting particle system on a finite torus tell us that it can be canonically decomposed in a gradient part, a circulation term and an harmonic component. All the three components can be computed and are associated with functions on the configuration space.