Brownian motion in one or more dimensions is extensively used as a stochastic process to model natural and engineering signals, as well as financial data. Most works dealing with multidimensional Brownian motion consider the different dimensions as independent components. In this article, we investigate a model of correlated Brownian motion in , where the individual components are not necessarily independent. We explore various statistical properties of the process under consideration, going beyond the conventional analysis of the second moment. Our particular focus lies on investigating the distribution of turning angles. This distribution reveals particularly interesting characteristics for processes with dependent components that are relevant to applications in diverse physical systems. Theoretical considerations are supported by numerical simulations and analysis of two real-world datasets: the financial data of the Dow Jones Industrial Average and the Standard and Poor’s 500, and trajectories of polystyrene beads in water. Finally, we show that the model can be readily extended to trajectories with correlations that change over time.
REFERENCES
1.
A.
Einstein
, “Über die von der
molekularkinetischen Theorie der Wärme geforderte Bewegung von in ruhenden
Flüssigkeiten suspendierten Teilchen
,” Ann.
Phys.
322
, 549
(1905
). 2.
M.
von Smoluchowski
, “Zur
kinetischen Theorie der Brownschen Molekularbewegung und der
Suspensionen
,” Ann. Phys.
21
, 756
(1906
). 3.
H.
Mori
, “Transport, collective
motion, and Brownian motion
,” Prog. Theor.
Phys.
33
, 423
–455
(1965
).
4.
P.
Mörters
and Y.
Peres
, Brownian
Motion
(Cambridge University
Press
, New York
,
2010
). Vol. 30.5.
P.
Hänggi
and F.
Marchesoni
, “Introduction:
100 years of Brownian motion
,” Chaos
15
, 026101
(2005
).6.
P.
Saffman
and M.
Delbrück
, “Brownian motion
in biological membranes
,” Proc. Natl. Acad. Sci.
U.S.A.
72
, 3111
–3113
(1975
). 7.
L. J.
Allen
, An Introduction to
Stochastic Processes with Applications to Biology
(CRC Press
, Boca
Raton
, 2010
).8.
W.
Ebeling
and I. M.
Sokolov
, Statistical
Thermodynamics and Stochastic Theory of Nonequilibrium
Systems
(World
Scientific
, Singapore
,
2005
).9.
P.
Hänggi
, P.
Talkner
, and M.
Borkovec
, “Reaction-rate
theory: Fifty years after Kramers
,” Rev. Mod.
Phys.
62
, 251
(1990
).10.
J.
Schwinger
, “Brownian motion
of a quantum oscillator
,” J. Math. Phys.
2
, 407
–432
(1961
).
11.
H.
Friedman
, D. A.
Kessler
, and E.
Barkai
, “Quantum walks: The
first detected passage time problem
,” Phys. Rev.
E
95
, 032141
(2017
). 12.
J. S.
Horne
, E. O.
Garton
, S. M.
Krone
, and J. S.
Lewis
, “Analyzing animal
movements using Brownian bridges
,” Ecology
88
, 2354
–2363
(2007
). 13.
O.
Vilk
, E.
Aghion
, T.
Avgar
, C.
Beta
, O.
Nagel
, A.
Sabri
, R.
Sarfati
, D. K.
Schwartz
, M.
Weiss
, D.
Krapf
et al. “Unravelling the origins of anomalous
diffusion: From molecules to migrating storks
,”
Phys. Rev. Res.
4
, 033055
(2022
). 14.
B. D.
Meyer
and H. M.
Saley
, “On the strategic
origin of Brownian motion in finance
,” Int. J. Game
Theory
31
, 285
–319
(2003
).
15.
A.
Borhani
and M.
Pätzold
, “Modelling of non-stationary
mobile radio channels using two-dimensional Brownian motion processes,” in
2013 International Conference on Advanced Technologies
for Communications (ATC 2013)
(IEEE, 2013), pp.
241–246.16.
L.
Shi
and R.
Wu
, “A probabilistic conflict detection
algorithm in terminal area based on three-dimensional Brownian motion,” in
2012 IEEE 11th International Conference on Signal
Processing (IEEE, 2012), Vol. 3, pp. 2287–2291.17.
X.
Gao
, “Image encryption
algorithm based on 2D hyperchaotic map
,” Opt. Laser
Technol.
142
, 107252
(2021
). 18.
M.
Rawat
, A. S.
Bafila
, S.
Kumar
, M.
Kumar
, A.
Pundir
, and S.
Singh
, “A new encryption
model for multimedia content using two dimensional Brownian motion and
coupled map lattice
,” Mult. Tools Appl.
82
, 43421
–43453
(2023
). 19.
K.
Giesecke
, “Correlated
default with incomplete information
,” J. Bank.
Financ.
28
, 1521
–1545
(2004
). 20.
W.-K.
Ching
, J.-W.
Gu
, and H.
Zheng
, “On correlated
defaults and incomplete information
,” J. Ind. Manag.
Optim.
17
, 889
–908
(2021
).21.
P. A.
Hassan
, S.
Rana
, and G.
Verma
, “Making sense of
Brownian motion: Colloid characterization by dynamic light
scattering
,” Langmuir
31
, 3
–12
(2015
).
22.
J. D.
Knight
and J. J.
Falke
, “Single-molecule
fluorescence studies of a PH domain: New insights into the membrane docking
reaction
,” Biophys. J.
96
, 566
–582
(2009
).
23.
G.
Campagnola
, K.
Nepal
, B. W.
Schroder
, O. B.
Peersen
, and D.
Krapf
, “Superdiffusive
motion of membrane-targeting C2 domains
,” Sci.
Rep.
5
, 17721
(2015
). 24.
D.
Krapf
, G.
Campagnola
, K.
Nepal
, and O. B.
Peersen
, “Strange kinetics
of bulk-mediated diffusion on lipid bilayers
,” Phys.
Chem. Chem. Phys.
18
, 12633
–12641
(2016
). 25.
D.
Krapf
, E.
Marinari
, R.
Metzler
, G.
Oshanin
, X.
Xu
, and A.
Squarcini
, “Power spectral
density of a single Brownian trajectory: What one can and cannot learn from
it
,” New J. Phys.
20
, 023029
(2018
). 26.
S.
Kou
and H.
Zhong
, “First-passage times
of two-dimensional Brownian motion
,” Adv. Appl.
Probab.
48
, 1045
–1060
(2016
). 27.
U.
Basu
, S. N.
Majumdar
, A.
Rosso
, and G.
Schehr
, “Active Brownian
motion in two dimensions
,” Phys. Rev. E
98
, 062121
(2018
). 28.
Ł.
Bielak
, A.
Grzesiek
, J.
Janczura
, and A.
Wyłomańska
, “Market risk
factors analysis for an international mining company. Multi-dimensional,
heavy-tailed-based modelling
,” Res. Policy
74
, 102308
(2021
). 29.
Y.
Shen
, G. B.
Giannakis
, and B.
Baingana
, “Nonlinear
structural vector autoregressive models with application to directed brain
networks
,” IEEE Trans. Signal Process.
67
, 5325
–5339
(2019
). 30.
D. B.
Mayer
, E.
Sarmiento-Gómez
, M. A.
Escobedo-Sánchez
, J. P.
Segovia-Gutiérrez
, C.
Kurzthaler
, S. U.
Egelhaaf
, and T.
Franosch
, “Two-dimensional
Brownian motion of anisotropic dimers
,” Phys. Rev.
E
104
, 014605
(2021
). 31.
O. M.
Maragó
, F.
Bonaccorso
, R.
Saija
, G.
Privitera
, P. G.
Gucciardi
, M. A.
Iatì
, G.
Calogero
, P. H.
Jones
, F.
Borghese
, P.
Denti
et al. “Brownian motion of
graphene
,” ACS Nano
4
, 7515
–7523
(2010
).32.
E.
Grong
and S.
Sommer
, “Most probable paths
for anisotropic Brownian motions on manifolds
,”
Found. Comput. Math.
24
, 313
–345
(2024
).
33.
D. A.
Tsyboulski
, S. M.
Bachilo
, A. B.
Kolomeisky
, and R. B.
Weisman
, “Translational and
rotational dynamics of individual single-walled carbon nanotubes in aqueous
suspension
,” ACS Nano
2
, 1770
–1776
(2008
).
34.
N.
Fakhri
, F. C.
MacKintosh
, B.
Lounis
, L.
Cognet
, and M.
Pasquali
, “Brownian motion
of stiff filaments in a crowded environment
,”
Science
330
, 1804
–1807
(2010
). 35.
H.
Föllmer
and P.
Protter
, “On Itô’s formula
for multidimensional Brownian motion
,” Prob. Theory
Relat. Fields
116
, 1
–20
(2000
).36.
J.
Mrongowius
and A.
Rößler
, “On the
approximation and simulation of iterated stochastic integrals and the
corresponding Lévy areas in terms of a multidimensional Brownian
motion
,” Stoch. Anal. Appl.
40
, 397
–425
(2022
).
37.
W. S.
Kendall
, “Coupling all the
Lévy stochastic areas of multidimensional Brownian motion
,”
Ann. Prob.
35
, 935
–953
(2007
).
38.
H.-H.
Kuo
and N.-R.
Shieh
, “A generalized Itô’s
formula for multidimensional Brownian motions and its
applications
,” Chin. J. Math.
15
, 163
–174
(1987
).39.
S.
Kou
and H.
Zhong
, “First-passage times
of two-dimensional Brownian motion
,” Adv. Appl.
Prob.
48
, 1045
–1060
(2016
). 40.
R.
Tsekov
and E.
Ruckenstein
,
“Two-dimensional Brownian motion of atoms and dimers on solid
surfaces
,” Surf. Sci.
344
, 175
–181
(1995
).
41.
L.
Sacerdote
, M.
Tamborrino
, and C.
Zucca
, “First passage times
of two-dimensional correlated processes: Analytical results for the Wiener
process and a numerical method for diffusion processes
,”
J. Comput. Appl. Math.
296
, 275
–292
(2016
).
42.
B.
Kopytko
and M.
Portenko
, “On a
multidimensional Brownian motion with a membrane located on a given
hyperplane
,” Stoch. Process. Appl.
160
, 371
–385
(2023
).
43.
J.
Blanchet
and K.
Murthy
, “Exact simulation of
multidimensional reflected Brownian motion
,” J.
Appl. Prob.
55
, 137
–156
(2018
).
44.
R.
Sant’Ana
, R.
Coelho
, and A.
Alcaim
, “Text-independent
speaker recognition based on the Hurst parameter and the multidimensional
fractional Brownian motion model
,” IEEE Trans. Audio
Speech Lang. Process.
14
, 931
–940
(2006
).45.
H.
Qian
, G. M.
Raymond
, and J. B.
Bassingthwaighte
, “On
two-dimensional fractional Brownian motion and fractional Brownian random
field
,” J. Phys. A: Math. Gen.
31
, L527
(1998
). 46.
K.
Maraj-Zygma̧t
, A.
Grzesiek
, G.
Sikora
, J.
Gajda
, and A.
Wyłomańska
, “Testing of
two-dimensional Gaussian processes by sample cross-covariance
function
,” Chaos
33
, 073135
(2023
).47.
L.
Ji
and S.
Robert
, “Ruin problem of a
two-dimensional fractional Brownian motion risk process
,”
Stoch. Models
34
, 73
–97
(2018
).
48.
D. R.
McGaughey
and G.
Aitken
, “Generating
two-dimensional fractional Brownian motion using the fractional Gaussian
process (FGp) algorithm
,” Phys. A
311
, 369
–380
(2002
).
49.
D. A. I.
Maruddani
and Trimono
,
“Modeling stock prices in a portfolio using multidimensional
geometric Brownian motion
,” J. Phys.: Conf.
Ser.
1025
, 012122
(2018
).50.
M.
Dominé
and V.
Pieper
, “First passage time
distribution of a two-dimensional Wiener process with
drift
,” Probab. Eng. Inform. Sci.
7
, 545
–555
(1993
).51.
L.
Sacerdote
and C.
Zucca
, “Joint distribution
of first exit times of a two dimensional Wiener process with jumps with
application to a pair of coupled neurons
,” Math.
Biosci.
245
, 61
–69
(2013
).
52.
P. J.
Thomas
and B.
Lindner
, “Phase descriptions
of a multidimensional Ornstein-Uhlenbeck process
,”
Phys. Rev. E
99
, 062221
(2019
). 53.
A.
Di Crescenzo
, V.
Giorno
, A. G.
Nobile
, and S.
Spina
, “First-exit-time
problems for two-dimensional Wiener and Ornstein–Uhlenbeck processes through
time-varying ellipses
,” Stochastics
96
, 696
–727
(2024
).
54.
G. H.
Golub
and C. F.
Van Loan
, Matrix
Computations
(JHU
Press
, 2013
).55.
H.
Qian
, M. P.
Sheetz
, and E. L.
Elson
, “Single particle
tracking. analysis of diffusion and flow in two-dimensional
systems
,” Biophys. J.
60
, 910
–921
(1991
).
56.
I.
Karatzas
and S.
Shreve
, Brownian Motion and
Stochastic Calculus
(Springer
, 2014
). Vol.
113.57.
F.
Wang
and A. E.
Gelfand
, “Modeling space and
space-time directional data using projected Gaussian
processes
,” J. Am. Stat. Assoc.
109
, 1565
–1580
(2014
). 58.
D.
Hernandez-Stumpfhauser
, F. J.
Breidt
, and M. J.
van der Woerd
, “The general
projected normal distribution of arbitrary dimension: Modeling and Bayesian
inference
,” Bayesian Anal.
12
, 113
(2017
). 59.
E. A.
Codling
, M. J.
Plank
, and S.
Benhamou
, “Random walk
models in biology
,” J. R. Soc. Interface
5
, 813
–834
(2008
).
60.
S.
Burov
, S. A.
Tabei
, T.
Huynh
, M. P.
Murrell
, L. H.
Philipson
, S. A.
Rice
, M. L.
Gardel
, N. F.
Scherer
, and A. R.
Dinner
, “Distribution of
directional change as a signature of complex dynamics
,”
Proc. Natl. Acad. Sci. U.S.A.
110
, 19689
–19694
(2013
). 61.
W. J.
Bos
, B.
Kadoch
, and K.
Schneider
, “Angular
statistics of Lagrangian trajectories in turbulence
,”
Phys. Rev. Lett.
114
, 214502
(2015
). 62.
S.
Sadegh
, J. L.
Higgins
, P. C.
Mannion
, M. M.
Tamkun
, and D.
Krapf
, “Plasma membrane is
compartmentalized by a self-similar cortical actin
meshwork
,” Phys. Rev. X
7
, 011031
(2017
).63.
B.
Kadoch
, W. J.
Bos
, and K.
Schneider
, “Directional
change of fluid particles in two-dimensional turbulence and of football
players
,” Phys. Rev. Fluids
2
, 064604
(2017
). 64.
A.
Mosqueira
, P. A.
Camino
, and F. J.
Barrantes
, “Antibody-induced
crosslinking and cholesterol-sensitive, anomalous diffusion of nicotinic
acetylcholine receptors
,” J. Neurochem.
152
, 663
–674
(2020
).
65.
L.
Fang
, L.
Li
, J.
Guo
, Y.
Liu
, and X.
Huang
, “Time scale of
directional change of active Brownian particles
,”
Phys. Lett. A
427
, 127934
(2022
). 66.
M.
Hidalgo-Soria
, Y.
Haddad
, E.
Barkai
, Y.
Garini
, and S.
Burov
, “Directed motion and spatial
coherence in the cell nucleus,” arXiv:2407.08899
(2024).67.
S.
Thapa
, A.
Wyłomańska
, G.
Sikora
, C. E.
Wagner
, D.
Krapf
, H.
Kantz
, A. V.
Chechkin
, and R.
Metzler
, “Leveraging
large-deviation statistics to decipher the stochastic properties of measured
trajectories
,” New J. Phys.
23
, 013008
(2021
). 68.
D.
Montiel
, H.
Cang
, and H.
Yang
, “Quantitative
characterization of changes in dynamical behavior for single-particle
tracking studies
,” J. Phys. Chem. B
110
, 19763
–19770
(2006
). 69.
T.
Akimoto
and E.
Yamamoto
, “Detection of
transition times from single-particle-tracking
trajectories
,” Phys. Rev. E
96
, 052138
(2017
). 70.
A.
Pacheco-Pozo
and D.
Krapf
, “Fractional Brownian
motion with fluctuating diffusivities
,” Phys. Rev.
E
110
, 014105
(2024
). 71.
A.
Pacheco-Pozo
, M.
Balcerek
, A.
Wyłomanska
, K.
Burnecki
, I. M.
Sokolov
, and D.
Krapf
, “Langevin equation in
heterogeneous landscapes: How to choose the interpretation
,”
Phys. Rev. Lett.
133
, 067102
(2024
). 72.
A.
Weron
, K.
Burnecki
, E. J.
Akin
, L.
Solé
, M.
Balcerek
, M. M.
Tamkun
, and D.
Krapf
, “Ergodicity breaking
on the neuronal surface emerges from random switching between diffusive
states
,” Sci. Rep.
7
, 5404
(2017
). 73.
G.
Muñoz-Gil
, H.
Bachimanchi
, J.
Pineda
, B.
Midtvedt
, M.
Lewenstein
, R.
Metzler
, D.
Krapf
, G.
Volpe
, and C.
Manzo
, “Quantitative evaluation of methods
to analyze motion changes in single-particle experiments,” arXiv:2311.18100 (2023).74.
F.
Lavancier
, A.
Philippe
, and D.
Surgailis
, “Covariance
function of vector self-similar processes
,” Stat.
Probab. Lett.
79
, 2415
–2421
(2009
). 75.
P.-O.
Amblard
and J.-F.
Coeurjolly
, “Identification
of the multivariate fractional Brownian motion
,”
IEEE Trans. Signal Process.
59
, 5152
–5168
(2011
). 76.
D.
Krapf
, N.
Lukat
, E.
Marinari
, R.
Metzler
, G.
Oshanin
, C.
Selhuber-Unkel
, A.
Squarcini
, L.
Stadler
, M.
Weiss
, and X.
Xu
, “Spectral content of a
single non-Brownian trajectory
,” Phys. Rev.
X
9
, 011019
(2019
).77.
A. V.
Chechkin
, V. Y.
Gonchar
, J.
Klafter
, and R.
Metzler
, “Fundamentals of Lévy flight
processes,” in Fractals, Diffusion, and Relaxation in Disordered Complex
Systems, Advances in Chemical Physics, Part B (Wiley, 2006), pp.
439–496.78.
V. V.
Palyulin
, G.
Blackburn
, M. A.
Lomholt
, N. W.
Watkins
, R.
Metzler
, R.
Klages
, and A. V.
Chechkin
, “First passage and
first hitting times of Lévy flights and Lévy walks
,”
New J. Phys.
21
, 103028
(2019
). 79.
K.
Burnecki
and A.
Weron
, “Fractional Lévy
stable motion can model subdiffusive dynamics
,”
Phys. Rev. E
82
, 021130
(2010
). 80.
J.
Janczura
, K.
Burnecki
, M.
Muszkieta
, A.
Stanislavsky
, and A.
Weron
, “Classification of
random trajectories based on the fractional Lévy stable
motion
,” Chaos, Solitons Fractals
154
, 111606
(2022
). 81.
A.
Ayache
and J.
Lévy Véhel
, “On the
identification of the pointwise Hölder exponent of the generalized
multifractional Brownian motion
,” Stoch. Process.
Their Appl.
111
, 119
–156
(2004
).
82.
W.
Wang
, M.
Balcerek
, K.
Burnecki
, A. V.
Chechkin
, S.
Janušonis
, J.
Śęzak
, T.
Vojta
, A.
Wyłomańska
, and R.
Metzler
,
“Memory-multi-fractional Brownian motion with continuous
correlations
,” Phys. Rev. Res.
5
, L032025
(2023
). 83.
J.
Ślęzak
and R.
Metzler
, “Minimal model of
diffusion with time changing Hurst exponent
,” J.
Phys. A: Math. Theor.
56
, 35LT01
(2023
).84.
M.
Balcerek
, A.
Wyłomańska
, K.
Burnecki
, R.
Metzler
, and D.
Krapf
, “Modelling
intermittent anomalous diffusion with switching fractional Brownian
motion
,” New J. Phys.
25
, 103031
(2023
). 85.
A. V.
Weigel
, B.
Simon
, M. M.
Tamkun
, and D.
Krapf
, “Ergodic and
nonergodic processes coexist in the plasma membrane as observed by
single-molecule tracking
,” Proc. Natl. Acad. Sci.
U.S.A.
108
, 6438
–6443
(2011
). 86.
R.
Kutner
and J.
Masoliver
, “The continuous
time random walk, still trendy: Fifty-year history, state of art and
outlook
,” Eur. Phys. J. B
90
, 50
(2017
). 87.
G.
Samorodnitsky
and M.
Taqqu
, Stable non-Gaussian
Random Processes
(Chapman &
Hall
, New York
,
1994
).© 2025 Author(s). Published under an exclusive license by AIP
Publishing.
2025
Author(s)
You do not currently have access to this content.
