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 $\log \log 1/P(W \leq x) \sim \gamma \log \log 1/x$ as $x \to 0$ with $\gamma > 1$, then the Laplace transform of $Y$ 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 $\Prob(Y \leq x)$ 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.