  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.
