Abstract.
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.
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The work was partially supported by NSF Grant DMS-0140405
Mathematics Subject Classification (2000): 60G17, 35K05, 60H15
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Krylov, N. One more square root law for Brownian motion and its application to SPDEs. Probab. Theory Relat. Fields 127, 496–512 (2003). https://doi.org/10.1007/s00440-003-0301-3
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DOI: https://doi.org/10.1007/s00440-003-0301-3