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

Abstract

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.Mathematics Subject Classification (2010). Primary 60E07, 60G51, 60G57; Secondary 28A80, 60G15, 81T99.

A review of conjectured laws of total mass of Bacry–Muzy GMC measures on the interval and circle and their applications

“…We also note that the intermittency differentiation approach is not limited to 1D or Bacry-Muzy measures or even GMC measures but in fact applies to all infinitely divisible multiplicative chaos measures, cf. [81]. 1 Hardy's method produces expressions for the logarithm of the Mellin transform in the integral form, which are more cumbersome than the expressions for the Mellin transform itself in terms of Barnes double gamma factors that are produced by the method of Fyodorov et.…”

“…To state this precisely, we need to briefly remind the reader of the key results of the intermittency renormalization formalism, cf. [69], [70], [74], [79]. Let us write the total mass in the form 16) and introduce the quantityφ…”

“…The answer depends on detailed, non-perturbative analysis of intermittency differentiation equations and requires novel mathematical tools. We refer the interested reader to [79] for an in-depth discussion of theoretical aspects of our approach. We also note that the intermittency differentiation approach is not limited to 1D or Bacry-Muzy measures or even GMC measures but in fact applies to all infinitely divisible multiplicative chaos measures, cf.…”

“…We also note that the intermittency differentiation approach is not limited to 1D or Bacry-Muzy measures or even GMC measures but in fact applies to all infinitely divisible multiplicative chaos measures, cf. [81].…”

“…While the positive integer moments of the total mass of all multiplicative chaos measures can be written in the form of multiple integrals, cf. [79], [80], the tractability of the Bacry-Muzy measures is inextricably tied to the fact that they are the only GMC measures that have explicitly known moments given by the Selberg integral on the interval and the Morris integral on the circle. A key challenge of the problem of computing the law of the total mass is that its moments diverge at any level of intermittency (inverse temperature) thereby rendering the associated Stieltjes moment problem indeterminate.…”