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We consider a Poisson process $\Phi$ on a general phase space. Theexpectation of a function of $\Phi$ can be considered as a functional of theintensity measure $\lambda$ of $\Phi$. Extending earlier results of Molchanovand Zuyev [Math. Oper. Res. 25 (2010) 485-508] on finite Poisson processes, westudy the behaviour of this functional under signed (possibly infinite)perturbations of $\lambda$. In particular, we obtain general Margulis-Russotype formulas for the derivative with respect to non-linear transformations ofthe intensity measure depending on some parameter. As an application, we studythe behaviour of expectations of functions of multivariate L\'evy processesunder perturbations of the L\'evy measure. A key ingredient of our approach isthe explicit Fock space representation obtained in Last and Penrose [Probab.Theory Related Fields 150 (2011) 663-690].

Perturbation analysis of Poisson processes