Logarithmic Potentials and Planar Brownian Motion
نویسنده
چکیده
In Section 5, we saw that for a Brownian motion process in n _ 3 dimensions, P (limtxIX, = o0) = 1 for all x. In sharp contrast to this situation, a planar Brownian motion is certain to hit any nonpolar set. THEOREM 8.1. Let B be a Borel set. Then PX(VB < cX) is either identically 1 or identically 0. PROOF. A simple computation shows that for any x e R2, 1' p(s, x) ds T co as t T oc. Thus, for any nonnegative function f having nonzero integral,
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