Brownian Motion: Local Distributions
This opaque title means “distributions related to local sample path properties of Brownian motion”. I have in mind properties such as Lévy’s estimate of the modulus of continuity, the corresponding results on small increments, the paradoxical fact that Brownian motion has local maxima but not points of increase, and self-intersection properties in d dimensions. Although these are “0–1” results, they can be regarded as consequences of stronger “distributional” assertions which can easily be derived via our heuristic. The topics of this section are more theoretical than were previous topics, though many are equivalent to more practical-looking problems on boundary-crossing.
KeywordsBrownian Motion Sojourn Time Local Distribution Small Increment Standard Brownian Motion
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