5,494 research outputs found
Unimodality for free L\'evy processes
We will prove that: (1) A symmetric free L\'evy process is unimodal if and
only if its free L\'evy measure is unimodal; (2) Every free L\'evy process with
boundedly supported L\'evy measure is unimodal in sufficiently large time. (2)
is completely different property from classical L\'evy processes. On the other
hand, we find a free L\'evy process such that its marginal distribution is not
unimodal for any time and its free L\'evy measure does not have a bounded
support. Therefore, we conclude that the boundedness of the support of free
L\'evy measure in (2) cannot be dropped. For the proof we will (almost)
characterize the existence of atoms and the existence of continuous probability
densities of marginal distributions of a free L\'evy process in terms of
L\'evy--Khintchine representation.Comment: 20 pages. To appear in Annales de l'Institut Henri Poincar\'e (B)
Probabilit\'es et Statistique
Gradient Estimate for Ornstein-Uhlenbeck Jump Processes
By using absolutely continuous lower bounds of the L\'evy measure, explicit
gradient estimates are derived for the semigroup of the corresponding L\'evy
process with a linear drift. A derivative formula is presented for the
conditional distribution of the process at time under the condition that
the process jumps before . Finally, by using bounded perturbations of the
L\'evy measure, the resulting gradient estimates are extended to linear SDEs
driven by L\'evy-type processes.Comment: 1
Limit theorems for L\'evy walks in dimensions: rare and bulk fluctuations
We consider super-diffusive L\'evy walks in dimensions when
the duration of a single step, i.e., a ballistic motion performed by a walker,
is governed by a power-law tailed distribution of infinite variance and finite
mean. We demonstrate that the probability density function (PDF) of the
coordinate of the random walker has two different scaling limits at large
times. One limit describes the bulk of the PDF. It is the dimensional
generalization of the one-dimensional L\'evy distribution and is the
counterpart of central limit theorem (CLT) for random walks with finite
dispersion. In contrast with the one-dimensional L\'evy distribution and the
CLT this distribution does not have universal shape. The PDF reflects
anisotropy of the single-step statistics however large the time is. The other
scaling limit, the so-called 'infinite density', describes the tail of the PDF
which determines second (dispersion) and higher moments of the PDF. This limit
repeats the angular structure of PDF of velocity in one step. Typical
realization of the walk consists of anomalous diffusive motion (described by
anisotropic dimensional L\'evy distribution) intermitted by long ballistic
flights (described by infinite density). The long flights are rare but due to
them the coordinate increases so much that their contribution determines the
dispersion. We illustrate the concept by considering two types of L\'evy walks,
with isotropic and anisotropic distributions of velocities. Furthermore, we
show that for isotropic but otherwise arbitrary velocity distribution the
dimensional process can be reduced to one-dimensional L\'evy walk
Randomly Weighted Self-normalized L\'evy Processes
Let be a bivariate L\'evy process, where is a subordinator
and is a L\'evy process formed by randomly weighting each jump of
by an independent random variable having cdf . We investigate the
asymptotic distribution of the self-normalized L\'evy process at 0
and at . We show that all subsequential limits of this ratio at 0
() are continuous for any nondegenerate with finite expectation if
and only if belongs to the centered Feller class at 0 (). We also
characterize when has a non-degenerate limit distribution at 0 and
.Comment: 32 page
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