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On exit laws for subordinated semigroups by means of 𝒞 1 -subordinators

Mohamed Hmissi, Ezzedine Mliki (2010)

Commentationes Mathematicae Universitatis Carolinae

We study the integral representation of potentials by exit laws in the framework of sub-Markovian semigroups of bounded operators acting on L 2 ( m ) . We mainly investigate subordinated semigroups in the Bochner sense by means of 𝒞 1 -subordinators. By considering the one-sided stable subordinators, we deduce an integral representation for the original semigroup.

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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