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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
    [tex]
    |B_t|=\sqrt{{B_t^{(1)}}^2+{B_t^{(2)}}^2}
    [/tex]
    which is a Bessel-2 process and defined
    [tex]
    T=\inf{t>0,|B_t|=1}
    [/tex]

    Then for any positive t
    [tex]
    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})\}]
    [/tex]

    where [itex]\gamma(z)[/itex] would be some kind of winding number:
    [tex]
    \gamma(B_t)=\int_0^t{dB_s\frac{1}{B_s}}
    [/tex]
    (I think w.p.1, a planar BM doesn't hit the origin so this should be fine.)

    I'd appreciate any input.

    Pere
     
    Last edited: Dec 18, 2008
  2. jcsd
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