Suppose that W(t) is a Wiener process in three dimension
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Suppose that W(t) is a Wiener process in three dimensions. Let w ? D, and define, as usual, p(w) = P(TH < TM | W(0) = w)
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Suppose that W(t) is a Wiener process in three dimensions. Let w ? D, and define, as usual, p(w) = P(TH < TM | W(0) = w). Let B be the ball of radius a with surface S, having area 4?a2 in three dimensions, where B ? D. Let TS be the first-passage time of the Wiener process W(t) from w to S. Show that P(TS < ?) = 1, and then use the strong Markov property at TS to deduce that 4?a2p(w) = x?S p(x) dS. Repeat the problem in two dimensions.
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