Variational Analysis of Poisson Processes

  • Ilya Molchanov
  • Sergei ZuyevEmail author
Part of the Bocconi & Springer Series book series (BS, volume 7)


The expected value of a functional F(η) of a Poisson process η can be considered as 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 \(F(\eta +\delta _{x}) - F(\eta )\)) 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 and optimal search. In non-asymptotic cases, it is in general possible to find the optimal measure using steepest descent algorithms which are based on the obtained explicit form of the gradient.


Poisson Process Point Process Cluster Centre Optimal Measure Poisson Point Process 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



The authors thank the organisers of the Oberwolfach Mini-Workshop: Stochastic Analysis for Poisson point processes (Feb. 2013) and the hospitality of the Oberwolfach Mathematical Institute for an excellent forum to exchange ideas partly presented in this paper.


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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Institute of Mathematical Statistics and Actuarial ScienceUniversity of BernBernSwitzerland
  2. 2.Department of Mathematical SciencesChalmers University of Technology and University of GothenburgGothenburgSweden

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