Exit distribution of MBM from a half-space and the Cauchy process (i) The MBM w(t) = (w 1 (t), w…

Exit distribution of MBM from a half-space and the Cauchy process (i) The MBM w(t) = (w1(t), w2(t), . . . , wd(t))in Rd starts in the upper half space, at w(0) = (0, 0, . . . , 0, z) with z > 0. Find the distribution of its exit points in the plane z = 0. (ii) Let τz be the FPT to the line z = 0 in R2 . Show that x(z) = w1(τz) is the Cauchy process, defined by the transition probability density function

 

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