Spectral expansion of the occupation measure for birth and death on a flow

A natural approach to investigating the dynamics of stochastic flows is to study the action of such flows on measures. In models of pollution transport the resulting measure-valued random process describes the distribution of pollutant mass in space. When, in addition, pollutants are allowed to enter and leave the space, the process is called “birth” and “death” on a flow. Here we study this model with an eye towards numerical calculation of its statistical descriptors (particularly the mean and covariance functionals). We propose a sequence of weak approximations convenient for numerics. This sequence is obtained by expanding a conditional mean measure in the eigenfunctions of a properly chosen self-adjoint operator.