Annales Henri Poincaré

, Volume 19, Issue 4, pp 1043–1079 | Cite as

A Theory of Intermittency Differentiation of 1D Infinitely Divisible Multiplicative Chaos Measures



A theory of intermittency differentiation is developed for a general class of 1D infinitely divisible multiplicative chaos measures. The intermittency invariance of the underlying infinitely divisible field is established and utilized to derive a Feynman–Kac equation for the distribution of the total mass of the limit measure by considering a stochastic flow in intermittency. The resulting equation prescribes the rule of intermittency differentiation for a general functional of the total mass and determines the distribution of the total mass and its dependence structure to the first order in intermittency. A class of non-local functionals of the limit measure extending the total mass is introduced and shown to be invariant under intermittency differentiation making the computation of the full high-temperature expansion of the total mass distribution possible in principle. For application, positive integer moments and covariance structure of the total mass are considered in detail.


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© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.StamfordUSA

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