This is an exerxise from G. Lawler's book on conformally invariant processes.(adsbygoogle = window.adsbygoogle || []).push({});

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 x^{a}.

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

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# Brownian motion

Can you offer guidance or do you also need help?

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