Combining multifractal additive and multiplicative chaos

The purpose of this article is the study of the new class of multifractal measures, which combines additive and multiplicative chaos, defined by nu(gamma,sigma) = Sigma(j >= 1) b(-j gamma)/j(2) Sigma(0 <= k <= bj-1) mu([kb(-j), (k+1)b(-j)))(sigma)delta(kb-j) (gamma >= 0, sigma >= 1), where mu is any positive Borel measure on [0, 1] and b is an integer >= 2. The singularities analysis of the measures nu(gamma,sigma) involves new results on the mass distribution of mu when mu describes large classes of multifractal measures. These results generalize ubiquity theorems associated with the Lebesgue measure. Under suitable assumptions on mu, the multifractal spectrum d(nu gamma,sigma) of nu(gamma,sigma) is linear on [0, h(gamma,sigma)] for some critical value h(gamma,sigma). Then d(nu gamma,sigma) is strictly concave on the right of h(gamma,sigma), and on this part it is deduced from the multifractal spectrum of mu by an affine transformation. This untypical shape is the result of the combination between Dirac masses and atomless multifractal measures. These measures satisfy multifractal formalisms. They open interesting perspectives in modeling discontinuous phenomena.