Small deviations in lognormal Mandelbrot cascades

We study small deviations in Mandelbrot cascades and some related models. Denoting by $Y$ the total mass variable of a Mandelbrot cascade generated by $W$, we show that if $\log \log 1\mathrm{/}P(W\le x)\sim \gamma \log \log 1\mathrm{/}x$ as $x \to 0$with$\gamma > 1$,thentheLaplacetransformof$Y$satisfies$\log \log 1/\E e^{-t Y} \sim \gamma \log \log t$as$t \to \infty$. As an application, this gives new estimates for $\text{Prob}(Y\le x)$ for small $x>0$. As another application of our methods, we prove a similar result for a variable arising as a total mass of a lognormal $\star$-scale invariant multiplicative chaos measure.