A solution to Schrödinger's problem of non-linear integral equations

A solution of Schrödinger's system of non-linear integral equations determines the rate function of a large deviation principle for kernels with prescribed marginal distributions. This kind of large deviation principle has some meaning in quantum mechanics.Diffusion equations associated with Schrödinger equations have typically transition functions with singular creation and killing. Hence they provide measurable non-negative generally unbounded kernels which may vanish on sets with positive measure and which can possess infinite mass.For Schrödinger systems with such kernels, a solution is proved to exist uniquely in terms of a product measure. It is obtained from a variational principle for the local adjoint of a product measure endomorphism. The generally unbounded factors of the solution are characterized by integrability properties.