Variational analysis of Poisson processes

The expected value of a functional F(η) of a Poisson process η can be considered a function of its intensity measure μ . The paper surveys several results concerning differentiability properties of this functional on the space of signed measures with finite total variation. Then necessary conditions for μ being a local minima of the considered functional are elaborated taking into account possible constraints on μ , most importantly the case of μ with given total mass a. These necessary conditions can be phrased by requiring that the gradient of the functional (being the expected first difference F(η + δx)−F(η)) is constant on the support of μ . In many important cases, the gradient depends only on the local structure of μ in a neighbourhood of x and so it is possible to work out the asymptotics of the minimising measure with the total mass a growing to infinity. Examples include the optimal approximation of convex functions, clustering problem, optimal search. In non-asymptotic cases, generally it is possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient.