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2 results on '"feynman graphs and rules"'

1. Markov random fields model and applications to image processing

Author

Boubaker Smii

Subjects
stochastic differential equations, lévy processes, markov random fields, gibbs distribution, feynman graphs and rules, Mathematics, QA1-939
Abstract

Markov random fields (MRFs) are well studied during the past 50 years. Their success are mainly due to their flexibility and to the fact that they gives raise to stochastic image models. In this work, we will consider a stochastic differential equation (SDE) driven by Lévy noise. We will show that the solution $ X_v $ of the SDE is a MRF satisfying the Markov property. We will prove that the Gibbs distribution of the process $ X_v $ can be represented graphically through Feynman graphs, which are defined as a set of cliques, then we will provide applications of MRFs in image processing where the image intensity at a particular location depends only on a neighborhood of pixels.

Published
2022
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2. The Feynman graph representation of convolution semigroups and its applications to Lévy statistics

Author

Boubaker Smii, Horst Thaler, and Hanno Gottschalk

Subjects
Statistics and Probability, Pure mathematics, Logarithm, convolution semigroups, Gaussian, Borel summability, Probabilistic logic, Feynman graph, Lévy distributions, maximum likelihood principle, symbols.namesake, Formalism (philosophy of mathematics), symbols, Feynman graphs and rules, Feynman diagram, Initial value problem, Mathematics - Probability, Mathematics
Abstract

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 L\'{e}vy distributions. Here, the Gaussian part of such a distribution is re-interpreted as a initial condition and a large diffusion expansion for L\'{e}vy densities is obtained. It is outlined how this expansion can be used in statistical problems that involve L\'{e}vy distributions., Comment: Published in at http://dx.doi.org/10.3150/07-BEJ106 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Published
2008

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