  This paper presents mathematical results in support of the methodology of the
probabilistic learning on manifolds (PLoM) recently introduced by the authors,
which has been used with success for analyzing complex engineering systems. The
PLoM considers a given initial dataset constituted of a small number of points
given in an Euclidean space, which are interpreted as independent realizations
of a vector-valued random variable for which its non-Gaussian probability
measure is unknown but is, \textit{a priori}, concentrated in an unknown subset
of the Euclidean space. The objective is to construct a learned dataset
constituted of additional realizations that allow the evaluation of converged
statistics. A transport of the probability measure estimated with the initial
dataset is done through a linear transformation constructed using a
reduced-order diffusion-maps basis. In this paper, it is proven that this
transported measure is a marginal distribution of the invariant measure of a
reduced-order It\^o stochastic differential equation that corresponds to a
dissipative Hamiltonian dynamical system. This construction allows for
preserving the concentration of the probability measure. This property is shown
by analyzing a distance between the random matrix constructed with the PLoM and
the matrix representing the initial dataset, as a function of the dimension of
the basis. It is further proven that this distance has a minimum for a
dimension of the reduced-order diffusion-maps basis that is strictly smaller
than the number of points in the initial dataset. Finally, a brief numerical
application illustrates the mathematical results.
