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CATEGORIES:Probability
SUMMARY:Unbiased Shifts for Brownian Motion - Hermann Thor
isson (University of Iceland)
DTSTART;TZID=Europe/London:20110926T141500
DTEND;TZID=Europe/London:20110926T151500
UID:TALK32521AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/32521
DESCRIPTION:Unbiased shifts of Brownian motion\nBased on joint
work with Günter Last and Peter Mörters\n\n\nLet
B = (B(t) : t in R) be a two-sided standard Brow
nian motion. Let T\nbe a real-valued measurable fu
nction of B. If T is a nonnegative stopping\ntime
then the shifted process (B(T + t) - B(T) : t no
nnegative) is a\none-sided Brownian motion indepe
ndent of B(T). However\, the two-sided\nprocess
(B(T + t) - B(T) : t in R) need 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\nCall T an unbiased shift
of B if (B(T + t) - B(T) : t in R) is a\nBrownia
n motion independent of B(T). Unbiased shifts can
be characterized\nin terms of allocation rules ba
lancing 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 p
roperties of unbiased shifts are given.\n\nThe cas
e when Q is concentrated at zero is of special int
erest. We obtain\na rigorous formulation of the in
tuitive idea that B looks globally the\nsame from
all its zeros\, thus resolving an issue raised by
Mandelbrot in\nThe Fractal Geometry of Nature. The
result can be stated as follows: if we\ntravel in
time according to the clock of local time we alwa
ys see a\ntwo-sided Brownian motion.\n\n\n
LOCATION:MR12\, CMS\, Wilberforce Road\, Cambridge\, CB3 0W
B
CONTACT:HoD Secretary\, DPMMS
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