Intersection Exponent for one-dimensional Brownian Motion












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We let $B^1,B^2$ be independent, one-dimensional Brownian Motions with $B^1(0)=1$ and $B^2(0)=-1$ and $T_n^i=inf{tgeq0:|B^i(t)|=n}$.



In Gregory Lawler's: Hausdorff Dimension of Cut-Points for Brownian Motion, it is claimed, that



$textbf{P}{B^1[0,T_n^1]cap B^2[0,T_n^2]=emptyset} approx n^{-2}$



where for functions $f$ and $g$, $fapprox g $ means $lim_{nto infty}frac{ln f(n)}{ln g(n)}=1$.



Now I don't really see why this holds. It might have to do something with the gambler's ruin problem, but the sets that we want to hit are random here ...
and I don't have more ideas on how to approach this.
Any help would be greatly appreciated!










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    We let $B^1,B^2$ be independent, one-dimensional Brownian Motions with $B^1(0)=1$ and $B^2(0)=-1$ and $T_n^i=inf{tgeq0:|B^i(t)|=n}$.



    In Gregory Lawler's: Hausdorff Dimension of Cut-Points for Brownian Motion, it is claimed, that



    $textbf{P}{B^1[0,T_n^1]cap B^2[0,T_n^2]=emptyset} approx n^{-2}$



    where for functions $f$ and $g$, $fapprox g $ means $lim_{nto infty}frac{ln f(n)}{ln g(n)}=1$.



    Now I don't really see why this holds. It might have to do something with the gambler's ruin problem, but the sets that we want to hit are random here ...
    and I don't have more ideas on how to approach this.
    Any help would be greatly appreciated!










    share|cite|improve this question



























      0












      0








      0







      We let $B^1,B^2$ be independent, one-dimensional Brownian Motions with $B^1(0)=1$ and $B^2(0)=-1$ and $T_n^i=inf{tgeq0:|B^i(t)|=n}$.



      In Gregory Lawler's: Hausdorff Dimension of Cut-Points for Brownian Motion, it is claimed, that



      $textbf{P}{B^1[0,T_n^1]cap B^2[0,T_n^2]=emptyset} approx n^{-2}$



      where for functions $f$ and $g$, $fapprox g $ means $lim_{nto infty}frac{ln f(n)}{ln g(n)}=1$.



      Now I don't really see why this holds. It might have to do something with the gambler's ruin problem, but the sets that we want to hit are random here ...
      and I don't have more ideas on how to approach this.
      Any help would be greatly appreciated!










      share|cite|improve this question















      We let $B^1,B^2$ be independent, one-dimensional Brownian Motions with $B^1(0)=1$ and $B^2(0)=-1$ and $T_n^i=inf{tgeq0:|B^i(t)|=n}$.



      In Gregory Lawler's: Hausdorff Dimension of Cut-Points for Brownian Motion, it is claimed, that



      $textbf{P}{B^1[0,T_n^1]cap B^2[0,T_n^2]=emptyset} approx n^{-2}$



      where for functions $f$ and $g$, $fapprox g $ means $lim_{nto infty}frac{ln f(n)}{ln g(n)}=1$.



      Now I don't really see why this holds. It might have to do something with the gambler's ruin problem, but the sets that we want to hit are random here ...
      and I don't have more ideas on how to approach this.
      Any help would be greatly appreciated!







      brownian-motion stopping-times






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      share|cite|improve this question













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      edited Dec 27 '18 at 13:42

























      asked Dec 24 '18 at 15:28









      John Doe

      163




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