One shows that the Stroock-Varadhan support theorem is valid in $\alpha$-Hölder norm. The central tool is an estimate of the probability that the Brownian motion has a large Hölder norm but a small uniform norm.
@article{hal-00120184,
author = {Ben Arous, G\'erard and Gradinaru, Mihai and Ledoux, Michel},
title = {H\"older norms and the support theorem for diffusions},
journal = {HAL},
volume = {1994},
number = {0},
year = {1994},
language = {en},
url = {http://dml.mathdoc.fr/item/hal-00120184}
}
Ben Arous, Gérard; Gradinaru, Mihai; Ledoux, Michel. Hölder norms and the support theorem for diffusions. HAL, Tome 1994 (1994) no. 0, . http://gdmltest.u-ga.fr/item/hal-00120184/