fractal and fractional
Article
Cesaro Limits for Fractional Dynamics
Yuri Kondratiev
1,2,†
and José da Silva
3,
*
,†
Citation: Kondratiev, Y.; da Silva, J.
Cesaro Limits for Fractional
Dynamics. Fractal Fract. 2021, 5, 133.
https://doi.org/10.3390/
fractalfract5040133
Academic Editor: Maja Andri´ c
Received: 25 August 2021
Accepted: 17 September 2021
Published: 22 September 2021
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1
Department of Mathematics, University of Bielefeld, D-33615 Bielefeld, Germany;
kondrat@math.uni-bielefeld.de
2
Institute of Mathematics of the National Academy of Sciences of Ukraine, National Pedagogical Dragomanov
University, 33615 Kiev, Ukraine
3
CIMA, University of Madeira, Campus da Penteada, 9020-105 Funchal, Portugal
* Correspondence: joses@staff.uma.pt; Tel.: +351-291-705-185
† These authors contributed equally to this work.
Abstract: We study the asymptotic behavior of random time changes of dynamical systems. As
random time changes we propose three classes which exhibits different patterns of asymptotic decays.
The subordination principle may be applied to study the asymptotic behavior of the random time
dynamical systems. It turns out that for the special case of stable subordinators explicit expressions
for the subordination are known and its asymptotic behavior are derived. For more general classes of
random time changes explicit calculations are essentially more complicated and we reduce our study
to the asymptotic behavior of the corresponding Cesaro limit.
Keywords: dynamical systems; random time change; inverse subordinator; asymptotic behavior
1. Introduction
In this paper we will deal with Markov processes or dynamical systems in R
d
. These
processes or dynamics starting from x ∈ R
d
, denote by X
x
(t), t ≥ 0, have associated
evolution equations on R
d
. In the Markov case we define for suitable f : R
d
−→ R the
function u(t, x)= E[ f ( X
x
(t))] which satisfied the Kolmogorov equation
∂
∂t
u(t, x)= Lu(t, x),
where L is the generator of the Markov process.
For a dynamical system we introduce u(t, x)= f ( X
x
(t)). Then this function is the
solution of the Liouville equation
∂
∂t
u(t, x)= Lu(t, x),
where now L is the Liouville operator for the dynamical system, see e.g., Kondratiev and
da Silva [1].
Let S(t), t ≥ 0 be a subordinator and E(t), t ≥ 0 denotes the inverse subordinator,
that is, for each t ≥ 0, E(t) := inf{s > 0 | S(s) > t}. This random process we consider as a
random time and assume to be independent of X
x
(t). Define a random process Y
x
by
Y
x
(t, ω) := X
x
( E(t, ω)).
Then as above we may introduce
u
E
(t, x)= E[ f (Y
x
(t))].
Fractal Fract. 2021, 5, 133. https://doi.org/10.3390/fractalfract5040133 https://www.mdpi.com/journal/fractalfract