Open Access
Translator Disclaimer
May 2008 The Feynman graph representation of convolution semigroups and its applications to Lévy statistics
Hanno Gottschalk, Boubaker Smii, Horst Thaler
Bernoulli 14(2): 322-351 (May 2008). DOI: 10.3150/07-BEJ106


We consider the Cauchy problem for a pseudo-differential operator which has a translation-invariant and analytic symbol. For a certain set of initial conditions, a formal solution is obtained by a perturbative expansion. The series so obtained can be re-expressed in terms of generalized Feynman graphs and Feynman rules. The logarithm of the solution can then be represented by a series containing only the connected Feynman graphs. Under some conditions, it is shown that the formal solution uniquely determines the real solution by means of Borel transforms. The formalism is then applied to probabilistic Lévy distributions. Here, the Gaussian part of such a distribution is re-interpreted as a initial condition and a large diffusion expansion for Lévy densities is obtained. It is outlined how this expansion can be used in statistical problems that involve Lévy distributions.


Download Citation

Hanno Gottschalk. Boubaker Smii. Horst Thaler. "The Feynman graph representation of convolution semigroups and its applications to Lévy statistics." Bernoulli 14 (2) 322 - 351, May 2008.


Published: May 2008
First available in Project Euclid: 22 April 2008

zbMATH: 1158.60361
MathSciNet: MR2544090
Digital Object Identifier: 10.3150/07-BEJ106

Keywords: Borel summability , convolution semigroups , Feynman graphs and rules , Lévy distributions , maximum likelihood principle

Rights: Copyright © 2008 Bernoulli Society for Mathematical Statistics and Probability


Vol.14 • No. 2 • May 2008
Back to Top