One more square root law for Brownian motion and its application to SPDEs

It is proved that there is a function p(c) ≥ 0 such that p(c) > 0 if c is large enough, and (a.s.) for any t ∈ [0, 1], the trajectory of Brownian motion after time t is contained in a parallel shift of the box [0, 2^-k] × [0, c2^-k/2] for all k belonging to a set with lower density ≥ p(c). This law of square root helps show that solutions of one-dimensional SPDEs are Hölder continuous up to the boundary.