In this paper we consider fractional Riesz–Bessel field of variable order, which is also known as multifractional Riesz–Bessel field. Sample path properties of this random field such as local regularity, locally self-similar property, Hausdorff dimension of the graph, and long∕short range dependent property are studied. The relationship between the multifractional Riesz–Bessel field and the multifractional Brownian field is also established.

1.
J. P.
Chiles
and
P.
Delfiner
,
Geostatistics: Modeling Spatial Uncertainty
(
Wiley
,
New York
,
1999
).
2.
G.
Christakos
,
Modern Spatiotemporal Geostatistics
(
Oxford University Press
,
New York
,
2000
).
3.
A. S.
Monin
and
A. M.
Yaglom
,
Statistical Fluid Mechanics: Mechanics of Turbulence
, Volume
2
(
MIT
,
Cambridge
,
1987
).
4.
U.
Frisch
,
Turbulence
(
Cambridge University Press
,
Cambridge
,
1995
).
6.
B.
Simon
,
The Euclidean (Quantum) Field Theory
(
Princeton University Press
,
Princeton
,
1974
).
7.
S.
Albeverio
,
H.
Gottschalk
, and
J. -L.
Wu
,
Rev. Math. Phys.
8
,
763
(
1996
).
8.
Applications of Fractional Calculus in Physics
, edited by
R.
Hilfer
(
World Scientific
,
River Edge, NJ
,
2000
).
9.
Y. L.
Kobelev
,
L. Y.
Kobelev
, and
Y. L.
Klimontovich
,
Dokl. Phys.
48
,
264
(
2003
).
10.
Y. L.
Kobelev
,
L. Y.
Kobelev
, and
Y. L.
Klimontovich
,
Dokl. Phys.
48
,
285
(
2003
).
11.
Y. L.
Kobelev
,
L. Y.
Kobelev
, and
Y. L.
Klimontovich
,
Dokl. Phys.
48
,
328
(
2003
).
12.
Y. L.
Kobelev
,
L. Y.
Kobelev
, and
E. P.
Romanov
,
Dokl. Phys.
48
,
409
(
2003
).
13.
A. V.
Popov
,
Russ. Phys. J.
48
,
947
(
2005
).
14.
R. F.
Bass
,
Probab. Theory Relat. Fields
79
,
271
(
1988
).
15.
M.
Tsuchiya
,
Stoch. Stoch. Rep.
38
,
95
(
1992
).
16.
K.
Kikuchi
and
A.
Negoro
,
Osaka J. Math.
34
,
319
(
1997
).
17.
N.
Jacob
and
H.-G.
Leopold
,
Integral Equ. Oper. Theory
17
,
544
(
1993
).
18.
W.
Hoh
,
Rev. Mat. Iberoam.
16
,
219
(
2000
).
19.
A.
Benassi
,
S.
Jaffard
, and
D.
Roux
,
Rev. Mat. Iberoam.
13
,
19
(
1997
).
20.
R.
Peltier
and
J.
Levy Vehel
, INRIA Report No. 2645,
1995
(unpublished).
21.
S. G.
Samko
and
B.
Ross
,
Integral Transforms Spec. Funct.
1
,
277
(
1993
).
22.
S. G.
Samko
,
J. Anal. Math.
21
,
213
(
1995
).
23.
L. Y.
Kobelev
, e-print arXiv:math.CA∕0002008v1.
24.
V. V.
Anh
,
J. M.
Angulo
, and
M. D.
Ruiz–Medina
,
J. Stat. Plan. Infer.
80
,
95
(
1999
).
25.
V. V.
Anh
,
J.
Maeda
,
Q. M.
Tieng
, and
H. T.
Tsui
,
Proceedings of International Conference on Image Processing
, ICIP 99,
1999
(unpublished), Vol.
9
, pp.
45
49
.
26.
S.
Deguy
,
C.
Debain
, and
A.
Benassi
, BMCV,
2000
(unpublished).
27.
T.
Lindstrom
,
Bull. London Math. Soc.
25
,
83
(
1993
).
28.
Z.
Ciesielski
and
A.
Kamont
,
Acta Sci. Math.
60
,
99
(
1995
).
29.
S.
Stoev
and
M. S.
Taqqu
,
Adv. Appl. Probab.
36
,
1085
(
2004
).
30.
S.
Stoev
and
M. S.
Taqqu
,
Fractals
13
,
157
(
2005
).
31.
A.
Ayache
,
J. Fourier Anal. Appl.
8
,
581
(
2002
).
32.
E.
Herbin
,
Rocky Mt. J. Math.
36
,
1249
(
2006
).
33.
C.
Lacaux
, Real harmonizable multifractional Levy motions,
Ann. Inst. Henri Poincare
40
,
259
(
2004
).
34.
C.
Lacaux
,
J. Theor. Probab.
18
,
481
(
2005
).
35.
M. D.
Ruiz–Medina
,
V. V.
Anh
, and
J. M.
Angulo
,
Stoch. Anal. Appl.
22
,
775
(
2004
).
36.
V. V.
Anh
,
J. M.
Angulo
, and
M. D.
Ruiz–Medina
,
Nonlinear Anal.: Real World Appl.
63
,
e2043
(
2005
).
37.
V. V.
Anh
,
M. D.
Ruiz-Medina
, and
J. M.
Angulo
,
Bull. Aust. Math. Soc.
62
,
319
(
2000
).
38.
V. V.
Anh
and
C. N.
Nguyen
,
Random Oper. Stochastic Equations
8
,
105
(
2000
).
39.
V. V.
Anh
,
C. C.
Heyde
, and
Q.
Tieng
,
J. Stat. Plan. Infer.
80
,
123
(
1999
).
40.
V. V.
Anh
,
W.
Grecksch
,
J. M.
Angulo
, and
M. D.
Ruiz-Medina
,
ANZIAM J.
42
,
41
(
2000
).
41.
V. V.
Anh
,
N. N.
Leonenko
, and
R.
McVinish
,
Fractals
9
,
329
(
2001
).
42.
V. V.
Anh
and
C. N.
Nguyen
,
Finance Stoch.
5
,
83
(
2001
).
43.
M. D.
Ruiz-Medina
,
J. M.
Angulo
, and
V. V.
Anh
,
J. Multivariate Anal.
85
,
192
(
2003
).
44.
L. D.
Pitt
and
R. S.
Robeva
,
Ann. Probab.
31
,
1338
(
2003
).
45.
S.
Bochner
,
Lectures on Fourier Integrals
(
Princeton University Press
,
Princeton, NJ
,
1959
).
46.
I. S.
Gradshteyn
and
I. M.
Ryzhik
,
Table of Integrals, Series, and Products
(
Academic
,
San Diego, CA
,
2000
).
47.
S. C.
Lim
and
C. H.
Eab
,
Phys. Lett. A
355
,
87
(
2006
).
48.
F.
Strocchi
,
Selected Topics on the General Properties of Quantum Field Theory
(
World Scientific
,
Singapore
,
1993
).
49.
R. S.
Robeva
and
L. D.
Pitt
,
Pliska Studia Mathematica
16
,
183
(
2004
).
50.
M. D.
Ruiz-Medina
,
J. M.
Angulo
, and
V. V.
Anh
,
Stoch. Anal. Appl.
21
,
465
(
2003
).
51.
M. D.
Ruiz-Medina
,
J. M.
Angulo
, and
V. V.
Anh
, Fractional random fields on domains with fractal boundary,
Infinite Dimen. Anal., Quantum Probab., Relat. Top.
7
,
395
(
2004
).
52.
A.
Ayache
and
S.
Leger
(unpublished).
53.
A.
Ayache
and
J.
Levy Vehel
,
Stat. Inference Stoch. Process.
3
,
7
(
2000
).
54.
K. J.
Falconer
,
J. Theor. Probab.
15
,
731
(
2002
).
55.
K. J.
Falconer
,
J. Lond. Math. Soc.
67
,
657
(
2003
).
56.
K. J.
Falconer
,
Fractal Geometry
(
Wiley
,
Hoboken, NJ
,
2003
).
57.
R. J.
Adler
,
The Geometry of Random Fields
(
Wiley
,
Chichester
,
1981
).
58.
Long-Range Dependence
, edited by
P.
Doukhan
,
G.
Oppenheim
, and
M. S.
Taqqu
(
Birkhäuser
,
Boston
,
2003
).
59.
G.
Rangarajan
and
M.
Ding
,
Processes with Long-Range Correlations: Theory and Applications
(
Springer
,
New York
,
2003
).
60.
V. V.
Anh
and
N. N.
Leonenko
,
J. Stat. Phys.
104
,
1349
(
2001
).
61.
V. V.
Anh
,
N. N.
Leonenko
, and
L. M.
Sakhno
,
J. Stat. Phys.
111
,
789
(
2003
).
62.
S.
Albeverio
,
H.
Gottschalk
, and
J.-L.
Wu
,
Commun. Math. Phys.
184
,
509
(
1997
).
63.
C. H.
Eab
,
S. C.
Lim
, and
L. P.
Teo
, “
Finite temperature Casimir effect for a massless fractional Klein-Gordon field with fractional Neumann conditions
,”
J. Math. Phys.
48
,
082301
(
2007
).
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