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SUMMARY:Unbiased Shifts for Brownian Motion - Hermann Thorisson  (Universi
 ty of Iceland)
DTSTART:20110926T131500Z
DTEND:20110926T141500Z
UID:TALK32521@talks.cam.ac.uk
CONTACT:HoD Secretary\, DPMMS
DESCRIPTION:Unbiased shifts of Brownian motion\nBased on joint work with G
 ünter Last and Peter Mörters\n\n\nLet  B = (B(t) : t in R)  be a two-sid
 ed standard Brownian motion. Let T\nbe a real-valued measurable function o
 f B.  If T is a nonnegative stopping\ntime then the shifted process  (B(T 
 + t) - B(T) : t nonnegative)  is a\none-sided Brownian motion independent 
 of B(T).  However\, the two-sided\nprocess  (B(T + t) - B(T) : t in R)  ne
 ed not be a Brownian motion.\nMoreover\, the example of a fixed time  T = 
 s  shows that even if it is\, it\nneed not be independent of  B(T).\n\nCal
 l T an unbiased shift of B if  (B(T + t) - B(T) : t in R)  is a\nBrownian 
 motion independent of B(T).  Unbiased shifts can be characterized\nin term
 s of allocation rules balancing additive functionals of B.  For any\nproba
 bility distribution Q on R we construct a stopping time T with the\nabove 
 properties such that B(T) has distribution Q.  Also moment and\nminimality
  properties of unbiased shifts are given.\n\nThe case when Q is concentrat
 ed at zero is of special interest. We obtain\na rigorous formulation of th
 e intuitive idea that B looks globally the\nsame from all its zeros\, thus
  resolving an issue raised by Mandelbrot in\nThe Fractal Geometry of Natur
 e. The result can be stated as follows: if we\ntravel in time according to
  the clock of local time we always see a\ntwo-sided Brownian motion.\n\n\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0WB
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