Page 560 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 560
DELING ROUGHNESS AND LONG-RANGE DEPENDENCE WITH FRACTIONAL
PROCESSES (MS-18)
Self-stabilizing processes
Jacques Levy Vehel, jacques.levy-vehel@inria.fr
Case Law Analytics, France
Coauthor: Kenneth Falconer
A self-stabilizing processes {Z(t), t ∈ [t0, t1)} is a random process which when localized, that
is scaled to a fine limit near a given t ∈ [t0, t1), has the distribution of an α(Z(t))-stable process,
where α : R → (0, 2) is a given continuous function. Thus the stability index near t depends on
the value of the process at t. In the case where α : R → (0, 1), we first construct deterministic
functions which satisfy a kind of autoregressive property involving sums over a plane point set
Π. Taking Π to be a Poisson point process then defines a random pure jump process, which we
show has the desired localized distributions.
When α may take values greater than 1, convergence of the considered sums may no longer
be absolute. We generalize the construction in two stages, firstly by setting up a process based
on a fixed point set but taking random signs of the summands, and then randomizing the point
set to get a process with the desired local properties.
Fractional integrals, derivatives and integral equations with weighted
Takagi-Landsberg functions
Vitalii Makogin, vitalii.makogin@uni-ulm.de
Ulm University, Germany
Coauthor: Yuliya Mishura
In the talk, we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg func-
tions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg func-
tions. Namely, for constants cm,k ∈ [−L, L], k, m ∈ N0, we define a weighted Takagi-
Landsberg function as yc,H : [0, 1] → R via
∞ 2m−1
yc,H (t) = 2m( 1 −H ) cm,kem,k(t), t ∈ [0, 1],
2
m=0 k=0
where H > 0, {em,k, m ∈ N0, k = 0, . . . , 2m − 1} are the Faber-Schauder functions on [0,1].
The class of the weighted Takagi-Landsberg functions of order H > 0 on [0, 1] coincides with
the H-Hölder continuous functions on [0, 1]. Based on computed fractional integrals and deriva-
tives of the Haar and Schauder functions, we get a new series representation of the fractional
derivatives of a Hölder continuous function. This result allows to get the new formula of a
Riemann-Stieltjes integral. The application of such series representation is the new method of
numerical solution of the Volterra and linear integral equations driven by a Hölder continuous
function.
558
PROCESSES (MS-18)
Self-stabilizing processes
Jacques Levy Vehel, jacques.levy-vehel@inria.fr
Case Law Analytics, France
Coauthor: Kenneth Falconer
A self-stabilizing processes {Z(t), t ∈ [t0, t1)} is a random process which when localized, that
is scaled to a fine limit near a given t ∈ [t0, t1), has the distribution of an α(Z(t))-stable process,
where α : R → (0, 2) is a given continuous function. Thus the stability index near t depends on
the value of the process at t. In the case where α : R → (0, 1), we first construct deterministic
functions which satisfy a kind of autoregressive property involving sums over a plane point set
Π. Taking Π to be a Poisson point process then defines a random pure jump process, which we
show has the desired localized distributions.
When α may take values greater than 1, convergence of the considered sums may no longer
be absolute. We generalize the construction in two stages, firstly by setting up a process based
on a fixed point set but taking random signs of the summands, and then randomizing the point
set to get a process with the desired local properties.
Fractional integrals, derivatives and integral equations with weighted
Takagi-Landsberg functions
Vitalii Makogin, vitalii.makogin@uni-ulm.de
Ulm University, Germany
Coauthor: Yuliya Mishura
In the talk, we find fractional Riemann-Liouville derivatives for the Takagi-Landsberg func-
tions. Moreover, we introduce their generalizations called weighted Takagi-Landsberg func-
tions. Namely, for constants cm,k ∈ [−L, L], k, m ∈ N0, we define a weighted Takagi-
Landsberg function as yc,H : [0, 1] → R via
∞ 2m−1
yc,H (t) = 2m( 1 −H ) cm,kem,k(t), t ∈ [0, 1],
2
m=0 k=0
where H > 0, {em,k, m ∈ N0, k = 0, . . . , 2m − 1} are the Faber-Schauder functions on [0,1].
The class of the weighted Takagi-Landsberg functions of order H > 0 on [0, 1] coincides with
the H-Hölder continuous functions on [0, 1]. Based on computed fractional integrals and deriva-
tives of the Haar and Schauder functions, we get a new series representation of the fractional
derivatives of a Hölder continuous function. This result allows to get the new formula of a
Riemann-Stieltjes integral. The application of such series representation is the new method of
numerical solution of the Volterra and linear integral equations driven by a Hölder continuous
function.
558
