On the Brownian curve and its circumscribing sphere (Q1092524)

This paper establishes the result that the two-dimensional Brownian trail \(\{\) B(t): \(0\leq t\leq 1\}\) has positive probability of touching its circumcircle in two distinct points. This contradicts an assertion due to P. Lévy. The proof depends on the following result. The Lebesgue measure of the planar set \(\{\) \(w: X-w,Y+w\) have each one point in common with the circumscribing circle of their union\(\}\) is positive. Here X, Y are general compact sets. Generalizations (for example to \(n>2\) dimensions) are indicated.