Conditional full support of Gaussian processes with stationary increments

We investigate the conditional full support (CFS) property, introduced in Guasoni et al. (2008a), for Gaussian processes with stationary increments. We give integrability conditions on the spectral measure of such a process which ensure that the process has CFS or not. In particular, the general results imply that, for a process with spectral density f such that $f(\lambda) \sim c_1 \lambda^p \rm{e}^{-c_2\lambda^q}$ for ¿ ¿ 8 (with necessarily p <1 if q = 0), the CFS property holds if and only if q <1.