Variational analysis for functionals of measures and of Poisson processes

Branches of science using measures are abound: length, volume, mass, electric charge, probability, point processes are all examples of measures. Looking at the set of measures as a convex cone in a Banach space of signed measures we derive specific first and second order necessary conditions for extrema of functionals of measures. Particularly nice form the corresponding derivatives take for functionals of a general Poisson point process, when their expectation is considered as a function of the intensity measure, making link to sensitivity analysis of Poisson driven stochastic systems, geometry of Poisson spaces and Malyavin calculus. We discuss applications in probability, statistics, optimal experimental design and numeric approximation of functions. The obtained explicit formulas for the gradient allows us to develop new steepest descent algorithms efficiency of which will be shown on numeric examples.