  Persistence diagrams play a fundamental role in Topological Data Analysis
where they are used as topological descriptors of filtrations built on top of
data. They consist in discrete multisets of points in the plane $\mathbb{R}^2$
that can equivalently be seen as discrete measures in $\mathbb{R}^2$. When the
data come as a random point cloud, these discrete measures become random
measures whose expectation is studied in this paper. First, we show that for a
wide class of filtrations, including the \v{C}ech and Rips-Vietoris
filtrations, the expected persistence diagram, that is a deterministic measure
on $\mathbb{R}^2$ , has a density with respect to the Lebesgue measure. Second,
building on the previous result we show that the persistence surface recently
introduced in [Adams & al., Persistence images: a stable vector representation
of persistent homology] can be seen as a kernel estimator of this density. We
propose a cross-validation scheme for selecting an optimal bandwidth, which is
proven to be a consistent procedure to estimate the density.
