Frédéric Chazal - Institut National de Recherche en Informatique et Automatique (INRIA) Persistence diagrams play a fundamental role in Topological Data Analysis (TDA) where they are used as topological descriptors of data represented as point cloud. They consist in discrete multisets of points in the plane that can equivalently be seen as discrete measures. When they are built on top of random data sets, persistence diagrams become random measures. In this talk, we will show that, in many cases, the expectation of these random discrete measures has a density with respect to the Lebesgue measure in the plane. We will discuss its estimation and show that various classical representations of persistence diagrams (persistence images, Betti curves,...) can be seen as kernel-based estimates of quantities deduced from it. This is a joint work with Vincent Divol (ENS Paris / Inria DataShape team).