  In this paper we answer the following question: what is the infinitesimal
generator of the diffusion process defined by a kernel that is normalized such
that it is bi-stochastic with respect to a specified measure? More precisely,
under the assumption that data is sampled from a Riemannian manifold we
determine how the resulting infinitesimal generator depends on the potentially
nonuniform distribution of the sample points, and the specified measure for the
bi-stochastic normalization. In a special case, we demonstrate a connection to
the heat kernel. We consider both the case where only a single data set is
given, and the case where a data set and a reference set are given. The
spectral theory of the constructed operators is studied, and Nystr\"om
extension formulas for the gradients of the eigenfunctions are computed.
Applications to discrete point sets and manifold learning are discussed.
