On the divergence of certain integrals of the Wiener process
Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 189-193.

Soit f une fonction non négative singulière seulement pour x=0 : f(x)=|x|-α, α>0. On étudie le comportement du processus de Wiener W(t) dans les voisinages à droite et à gauche des traversées d’un niveau, et on donne des conditions nécessaires et suffisantes pour que les intégrales de f(W(t)) soient finies ou infinies.

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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     author = {Shepp, Lawrence A. and Klauder, John R. and Ezawa, Hiroshi},
     title = {On the divergence of certain integrals of the {Wiener} process},
     journal = {Annales de l'Institut Fourier},
     pages = {189--193},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {24},
     number = {2},
     year = {1974},
     doi = {10.5802/aif.512},
     mrnumber = {53 #6772},
     zbl = {0275.60088},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.512/}
}
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Shepp, Lawrence A.; Klauder, John R.; Ezawa, Hiroshi. On the divergence of certain integrals of the Wiener process. Annales de l'Institut Fourier, Tome 24 (1974) no. 2, pp. 189-193. doi : 10.5802/aif.512. https://www.numdam.org/articles/10.5802/aif.512/

[EKS] H. Ezawa, J. R. Klauder and L. A. Shepp, Vestigial Effects of Singular Potentials in Diffusion Theory and Quantum Mechanics. J. Math. Phys., to appear.

[IMcK] K. Ito and H. P. Mckean, Diffusion Processes and Their Sample Paths, Springer (1965). | MR | Zbl

[KI] J. R. Klauder, Field Structure Through Model Studies: Aspects of Nonrenormalizable Theories, Acta Phys. Austriaca, Suppl. XI (1973), 341-387.

[Kn] F. B. Knight, Existence of Small Oscillations at Zeros of Brownian Motion, manuscript. | Numdam | Zbl

[R] D. Ray, Sojourn Times of Diffusion Processes, Ill. J. Math., 7 (1963), 615-630. | MR | Zbl

[S] L. A. Shepp, Radon-Nikodym Derivatives of Gaussian Measures, Ann. Math. Stat., 37 (1966), 321-354. | MR | Zbl

[V] D. E. Varberg, Equivalent Gaussian Measures with a Particularly Simple Radon-Nikodym Derivative, Ann. Math. Stat., 38 (1967), 1027-1030. | MR | Zbl

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