Vol. 60, No. 2, 1975

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Brownian motion and sets of multiplicity

Robert P. Kaufman

Vol. 60 (1975), No. 2, 111–114

X(t) is Brownian motion on the axis −∞ < t < , with paths in Rn, n 2. X(t) leads to composed mappings f X, where f is a real-valued function of class Λα(Rn), whose gradient never vanishes. To define the class Λα(Rn), when α > 1, take the integer p in the interval α 1 p < α and require that f have continuous partial derivatives of orders 1,,p and these fulfill a Lipschitz condition in exponent α p on each compact set; to specify further that grad f0 throughout Rn, write Λ+α. Then a closed set T is a set of “Λα-multiplicity” if every transform f(T) R1(f Λ+α) is a set of strict multiplicity— an M0-set (see below). Henceforth we define b = α1 and take S to be a closed linear set.

Mathematical Subject Classification 2000
Primary: 60J65
Secondary: 42A48
Received: 11 December 1973
Revised: 15 March 1974
Published: 1 October 1975
Robert P. Kaufman