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Brownian motion

  1. Dec 18, 2008 #1
    This is an exerxise from G. Lawler's book on conformally invariant processes.

    Show there exist constants 0<a,c<inf such that the probability p for a planar Brownian Motion B_t starting at x makes a loop around the origin before hitting the unit circle is at least 1-c xa.

    I have not really an idea how to show this. I considered
    which is a Bessel-2 process and defined

    Then for any positive t
    p\geq \mathbb{P}[\{T\geq t\}\cap \{\exists s,s'\in[0,t]:B_s=B_{s'} \text{ and } \gamma(B_s)-\gamma(B_{s'})\in \mathbb{Z})\}]

    where [itex]\gamma(z)[/itex] would be some kind of winding number:
    (I think w.p.1, a planar BM doesn't hit the origin so this should be fine.)

    I'd appreciate any input.

    Last edited: Dec 18, 2008
  2. jcsd
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