We prove strong small deviations results for Brownian motion under independent time-changes satisfying their own asymptotic criteria. We then apply these results to certain stochastic integrals which are elements of second-order homogeneous chaos.
"Small deviations for time-changed Brownian motions and applications to second-order chaos." Electron. J. Probab. 19 1 - 23, 2014. https://doi.org/10.1214/EJP.v19-2993