The link between Markovian and non-Markovian stochastic processes is examined by looking at the drift and diffusion coefficients. Starting with the Langevin equation, a solution for the Fokker-Planck equation is obtained using white noise analysis. An evaluation of the mean square displacement explicitly shows that the drift coefficient may not play a crucial role in transitions from Markovian to non-Markovian processes. A special case of the solution obtained for the Fokker-Planck equation is fractional Brownian motion which we use to consider absorbing boundaries.
REFERENCES
1.
B.-H.
Liu
, L.
Li
, Y.-F.
Huang
, C.-F.
Li
, G.-C.
Guo
, E.-M.
Laine
, H.-P.
Breuer
, and J.
Piilo
, “Experimental control of the transition from Markovian to non-Markovian dynamics of open quantum systems
,” Nat. Phys.
7
(2011
) 931
–934
.2.
R.
Metzler
, J. -H.
Jeon
, A. G.
Cherstvy
, and E.
Barkai
. “Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking
,” Phys. Chem. Chem. Phys.
16
(2014
) 24128
.3.
C. C.
Bernido
and M. V.
Carpio-Bernido
, Methods and Applications of White Noise Analysis in Interdisciplinary Sciences
, (World Scientific
, Singapore
, 2014
).4.
W.
Barredo
, C. C.
Bernido
, M. V.
Carpio-Bernido
, and J. B.
Bornales
, “Modelling non-Markovian fluctuations in intracellular biomolecular transport
.” Math. Biosci.
297
(2018
) 27
–31
.5.
R.
Aure
, C. C.
Bernido
, M. V.
Carpio-Bernido
, and R. G.
Bacabac
, “Damped white noise diffusion with memory for diffusing microprobes in ageing fibrin gels
,” Biophys. Jour.
117
(2019
) 1029
–1036
.6.
R. R.
Violanda
, C. C.
Bernido
, and M. V.
Carpio-Bernido
, “White noise functional integral for exponentially decaying memory: Nucleotide distribution in bacterial genomes
,” Phys. Scripta
94
(2019
) 125006
.7.
C. C.
Bernido
and M. V.
Carpio-Bernido
, “White noise analysis: some applications in complex systems, biophysics and quantum mechanics
,” Int. J. Mod. Phys. B
26
(2012
) 1230014
.8.
9.
T.
Hida
, H. H.
Kuo
, J.
Potthoff
, and L.
Streit
, White Noise. An Infinite Dimensional Calculus
(Kluwer
, Dordrecht
, 1993
).10.
A.
Molini
, P.
Talkner
, G. G.
Katul
, and A.
Porporato
, “First passage time statistics of Brownian motion with purely time dependent drift and diffusion
,” Phys. A: Stat. Mech. Appl.
390
(2011
) 1841
–1852
.11.
H.-H.
Kuo
, “Donsker's delta function as a generalized Brownian functional and its application,” Theory and Application of Random Fields, Lect. Notes Control Inf. Sci.
, Vol. 49
(Springer
, Berlin
, 1983
) 167
–78
.12.
A.
Lascheck
, P.
Leukert
, L.
Streit
, and W.
Westerkamp
, “More about Donsker's delta function
,” Soochow J. Math.
20
(1994
) 401
–18
.13.
I. S.
Gradshteyn
and I. M.
Ryzhik
, Table of Integrals, Series, and Products
(Academic Press
, 2014
).14.
I.
Calvo
and R.
Sanchez
, “The path integral formulation of fractional Brownian motion for the general Hurst exponent
.” Jour. Phys. A: Math. Theor.
41
(2008
) 282002
.15.
16.
K. S.
Fa
, “Solution of Fokker-Planck equation for a broad class of drift and diffusion coefficients
,” Phys. Rev. E
84
(2011
) 012102
.
This content is only available via PDF.
© 2020 Author(s).
2020
Author(s)
