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 (element of) [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 Holder continuous up to the boundary.