We consider the continuous model of log-infinitely divisible multifractal random measures (MRM) introduced in \cite{ bacry} . If M is a non degenerate multifractal measure with associated metric $\rho(x,y)=M([x,y])$ and structure function $\zet a$, we show that we have the following relation between the (Euclidian) Hausdorff dimension ${ \rm dim}_H$ of a measurable set K and the Hausdorff dimension ${ \rm dim}_H^{ \rho}$ with respect to \rho of the same set: $\zeta({ \rm dim}_H^{ \rho}(K))={ \r m dim}_H(K)$. Our results can be extended to higher dimensions in the log normal case: inspired by quantum gravity in dime nsion 2, we consider the 2 dimensional case.
Categories: math.PR