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On the divergence of certain integrals of the Wiener process

Lawrence A. Shepp; John R. Klauder; Hiroshi Ezawa — 1974

Annales de l'institut Fourier

Let $f (x)$ be a nonnegative function with its only singularity at $x = 0$ , e.g. $f (x) = | x |^{- α}$ , $α > 0$ . We study the behavior of the Wiener process $W (t)$ in left and right hand neighborhoods of level crossings by finding necessary and sufficient conditions on $f$ for the integrals of $f (W (t))$ to be finite or infinite.

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