13,850 research outputs found
L\'evy mixing related to distributed order calculus, subordinators and slow diffusions
The study of distributed order calculus usually concerns about fractional
derivatives of the form for some
measure , eventually a probability measure. In this paper an approach based
on L\'evy mixing is proposed. Non-decreasing L\'evy processes associated to
L\'evy triplets of the form \l a(y), b(y), \nu(ds, y) \r are considered and
the parameter is randomized by means of a probability measure. The related
subordinators are studied from different point of views. Some distributional
properties are obtained and the interplay with inverse local times of Markov
processes is explored. Distributed order integro-differential operators are
introduced and adopted in order to write explicitly the governing equations of
such processes. An application to slow diffusions is discussed.Comment: in Journal of Mathematical Analysis and Applications (2015
Fractional Calculus for Continuum Mechanics - anisotropic non-locality
In this paper the generalisation of previous author's formulation of
fractional continuum mechanics to the case of anisotropic non-locality is
presented. The considerations include the review of competitive formulations
available in literature. The overall concept bases on the fractional
deformation gradient which is non-local, as a consequence of fractional
derivative definition. The main advantage of the proposed formulation is its
analogical structure to the general framework of classical continuum mechanics.
In this sense, it allows, to give similar physical and geometrical meaning of
introduced objects
Non-local fractional model of rate independent plasticity
In the paper the generalisation of classical rate independent plasticity
using fractional calculus is presented. This new formulation is non-local due
to properties of applied fractional differential operator during definition of
kinematics. In the description small fractional strains assumption is hold
together with additive decomposition of total fractional strains into elastic
and plastic parts. Classical local rate independent plasticity is recovered as
a special case
Stochastic calculus for convoluted L\'{e}vy processes
We develop a stochastic calculus for processes which are built by convoluting
a pure jump, zero expectation L\'{e}vy process with a Volterra-type kernel.
This class of processes contains, for example, fractional L\'{e}vy processes as
studied by Marquardt [Bernoulli 12 (2006) 1090--1126.] The integral which we
introduce is a Skorokhod integral. Nonetheless, we avoid the technicalities
from Malliavin calculus and white noise analysis and give an elementary
definition based on expectations under change of measure. As a main result, we
derive an It\^{o} formula which separates the different contributions from the
memory due to the convolution and from the jumps.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ115 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Fractional and noncommutative spacetimes
We establish a mapping between fractional and noncommutative spacetimes in
configuration space. Depending on the scale at which the relation is
considered, there arise two possibilities. For a fractional spacetime with
log-oscillatory measure, the effective measure near the fundamental scale
determining the log-period coincides with the non-rotation-invariant but
cyclicity-preserving measure of \kappa-Minkowski. At scales larger than the
log-period, the fractional measure is averaged and becomes a power-law with
real exponent. This can be also regarded as the cyclicity-inducing measure in a
noncommutative spacetime defined by a certain nonlinear algebra of the
coordinates, which interpolates between \kappa-Minkowski and canonical
spacetime. These results are based upon a braiding formula valid for any
nonlinear algebra which can be mapped onto the Heisenberg algebra.Comment: 15 pages. v2: typos correcte
- …