Abstract

In this paper, the widely known path integral method, derived from the application of the Chapman–Kolmogorov equation, is described in details and discussed with reference to the main results available in literature in several decades of contributions. The most simple application of the method is related to the solution of Fokker–Planck type equations. In this paper, the solution in the presence of normal, α-stable, and Poissonian white noises is first discussed. Then, application to barrier problems, such as first passage problems and vibroimpact problems is described. Further, the extension of the path integral method to problems involving multi-degrees-of-freedom systems is analyzed. Lastly, an alternative approach to the path integration method, that is the Wiener Path integration (WPI), also based on the Chapman–Komogorov equation, is discussed. The main advantages and the drawbacks in using these two methods are deeply analyzed and the main results available in literature are highlighted.

Issue Section:
Review Article
Topics:
White noise, Path integrals, Nonlinear systems

References

1.
Einstein
,
A.
,
1905
, “
On the Movement of Small Particles Suspended in Stationary Liquids Required by the Molecular-Kinetic Theory of Heat (in German)
,”
Ann. Der Phys.
,
322
(
8
), pp.
549
–
560
.10.1002/andp.19053220806
2.
Brown
,
R.
,
1828
, “
XXVII: A Brief Account of Microscopical Observations Made in the Months of June, July and August 1827, on the Particles Contained in the Pollen of Plants; and on the General Existence of Active Molecules in Organic and Inorganic Bodies
,”
Philos. Mag.
,
4
(
21
), pp.
161
–
173
.10.1080/14786442808674769
3.
Fokker
,
A. D.
,
1914
, “
Die Mittlere Energie Rotierender Elektrischer Dipole im Strahlungsfeld
,”
Annalen Der Phys.
,
348
(
5
), pp.
810
–
820
.10.1002/andp.19143480507
4.
Planck
,
M.
,
1917
, “
Ber Einen Satz Der Statistischen Dynamik Und Seine Erweiterung in Der Quantentheorie
,”
Sitzungsberichte Der Preussischen Akademie Der Wissenschaften zu Berlin
,
24
, pp.
324
–
341
.
5.
Itô
,
K.
,
1941
, “
On Stochastic Processes (Infinitely Divisible Laws of Probability)
,”
Jpn. J. Math.
,
18
(
0
), pp.
261
–
301
.10.4099/jjm1924.18.0_261
6.
Itô
,
K.
,
1946
, “
On a Stochastic Integral Equation
,”
Proc. Jpn. Acad.
,
22
(
2
), pp.
32
–
35
.10.3792/pja/1195572371
7.
Itô
,
K.
,
1951
, “
On a Formula Concerning Stochastic Differentials
,”
Nagoya Math. J.
,
3
, pp.
55
–
75
.10.1017/S0027763000012216
8.
Itô
,
K.
,
1951
, “
On Stochastic Differential Equations
,”
Memoirs Am. Math. Soc.
,
4
, pp.
1
–
51
.
9.
Cornish
,
E. A.
, and
Fisher
,
R. A.
,
1938
, “
Moments and Cumulants in the Specification of Distributions
,”
Revue de L'Institut Int. de Statistique/Rev. Int. Stat. Inst.
,
5
(
4
), pp.
307
–
320
.10.2307/1400905
10.
Bover
,
D.
,
1978
, “
Moment Equation Methods for Nonlinear Stochastic Systems
,”
J. Math. Anal. Appl.
,
65
(
2
), pp.
306
–
320
.10.1016/0022-247X(78)90182-8
11.
Ibrahim
,
R.
,
1978
, “
Stationary Response of a Randomly Parametric Excited Nonlinear System
,”
ASME J. Appl. Mech.
,
45
(
4
), pp.
910
–
916
.10.1115/1.3424440
12.
Crandall
,
S.
,
1980
, “
Non-Gaussian Closure for Random Vibration of Non-Linear Oscillators
,”
Int. J. Non-Linear Mech.
,
15
(
4–5
), pp.
303
–
313
.10.1016/0020-7462(80)90015-3
13.
Ibrahim
,
R.
, and
Soundararajan
,
A.
,
1983
, “
Non-Linear Parametric Liquid Sloshing Under Wide Band Random Excitation
,”
J. Sound Vib.
,
91
(
1
), pp.
119
–
134
.10.1016/0022-460X(83)90455-8
14.
Wu
,
W.
, and
Lin
,
Y.
,
1984
, “
Cumulant-Neglect Closure for Non-Linear Oscillators Under Random Parametric and External Excitations
,”
Int. J. Non-Linear Mech.
,
19
(
4
), pp.
349
–
916
.10.1016/0020-7462(84)90063-5
15.
Ibrahim
,
R.
,
Soundararajan
,
A.
, and
Heo
,
H.
,
1985
, “
Stochastic Response of Nonlinear Dynamic Systems Based on a Non-Gaussian Closure
,”
ASME J. Appl. Mech.
,
52
(
4
), pp.
965
–
970
.10.1115/1.3169176
16.
Lutes
,
L.
,
1986
, “
State Space Analysis of Stochastic Response Cumulants
,”
Probab. Eng. Mech.
,
1
(
2
), pp.
94
–
98
.10.1016/0266-8920(86)90031-7
17.
Di Paola
,
M.
,
Falsone
,
G.
, and
Muscolino
,
G.
,
1990
, “
Random Analysis of Geometrically Non-Linear FE Modelled Structures Under Seismic Actions
,”
Struct. Saf.
,
8
(
1–4
), pp.
209
–
222
.10.1016/0167-4730(90)90041-M
18.
Grigoriu
,
M.
,
1991
, “
A Consistent Closure Method for Non-Linear Random Vibration
,”
Int. J. Non-Linear Mech.
,
26
(
6
), pp.
857
–
866
.10.1016/0020-7462(91)90037-T
19.
Lutes
,
L.
, and
Chen
,
D.
,
1992
, “
Stochastic Response Moments for Linear Systems
,”
Probab. Eng. Mech.
,
7
(
3
), pp.
165
–
173
.10.1016/0266-8920(92)90020-I
20.
Di Paola
,
M.
,
Falsone
,
G.
, and
Pirrotta
,
A.
,
1992
, “
Stochastic Response Analysis of Nonlinear Systems Under Gaussian Inputs
,”
Probab. Eng. Mech.
,
7
(
1
), pp.
15
–
21
.10.1016/0266-8920(92)90004-2
21.
Hasofer
,
A.
, and
Grigoriu
,
M.
,
1995
, “
A New Perspective on the Moment Closure Method
,”
ASME J. Appl. Mech.
,
62
(
2
), pp.
527
–
532
.10.1115/1.2895962
22.
Papadimitriou
,
C.
, and
Lutes
,
L.
,
1996
, “
Stochastic Cumulant Analysis of MDOF Systems With Polynomial-Type Nonlinearities
,”
Probab. Eng. Mech.
,
11
(
1
), pp.
1
–
13
.10.1016/0266-8920(95)00022-4
23.
Gullo
,
I.
,
Muscolino
,
G.
, and
Vasta
,
M.
,
1998
, “
Non-Gaussian Probability Density Function of SDOF Linear Structures Under Wind Actions
,”
J. Wind Eng. Ind. Aerodyn.
,
74–76
, pp.
1123
–
1134
.10.1016/S0167-6105(98)00103-2
24.
Benfratello
,
S.
,
Caddemi
,
S.
, and
Muscolino
,
G.
,
2000
, “
Gaussian and non-Gaussian Stochastic Sensitivity Analysis of Discrete Structural System
,”
Comput. Struct.
,
78
(
1–3
), pp.
425
–
434
.10.1016/S0045-7949(00)00086-9
25.
Grigoriu
,
M.
,
2008
, “
A Critical Evaluation of Closure Methods Via Two Simple Dynamic Systems
,”
J. Sound Vib.
,
317
(
1–2
), pp.
190
–
198
.10.1016/j.jsv.2008.02.049
26.
Lin
,
Y.
,
1967
,
Probabilistic Theory of Structural Dynamics
,
McGraw-Hill
, New York.
27.
Roberts
,
J.
, and
Spanos
,
P.
,
2003
,
Random Vibration and Statistical Linearization
(Dover Civil and Mechanical Engineering Series),
Dover Publications
, Mineola, NY.
28.
Feller
,
W.
,
1968
,
An Introduction to Probability Theory and Its Applications
,
Wiley
,
New York
.
29.
Karlin
,
S.
, and
Taylor
,
H.
,
1975
,
A First Course in Stochastic Processes
,
Academic Press
,
San Diego, CA
.
30.
Samorodnitsky
,
G.
, and
Taqqu
,
M.
,
1994
,
Stable Non-Gaussian Random Processes: Stochastic Models With Infinite Variance
(Stochastic Modeling Series),
Chapman & Hall/CRC
, New York.
31.
Hald
,
A.
,
2000
, “
The Early History of the Cumulants and the Gram-Charlier Series
,”
Int. Stat. Rev.
,
68
(
2
), pp.
137
–
153
.10.1111/j.1751-5823.2000.tb00318.x
32.
Cavaleri
,
L.
, and
Di Paola
,
M.
,
2000
, “
Statistic Moments of the Total Energy of Potential Systems and Application to Equivalent Non-Linearization
,”
Int. J. Non-Linear Mech.
,
35
(
4
), pp.
573
–
587
.10.1016/S0020-7462(99)00020-7
33.
Cavaleri
,
L.
,
Di Paola
,
M.
, and
Failla
,
G.
,
2003
, “
Some Properties of Multi-Degree-of-Freedom Potential Systems and Application to Statistical Equivalent Non-Linearization
,”
Int. J. Non-Linear Mech.
,
38
(
3
), pp.
405
–
421
.10.1016/S0020-7462(01)00080-4
34.
Cai
,
G.
, and
Lin
,
Y.
,
1988
, “
On Exact Stationary Solutions of Equivalent Non-Linear Stochastic Systems
,”
Int. J. Non-Linear Mech.
,
23
(
4
), pp.
315
–
325
.10.1016/0020-7462(88)90028-5
35.
Cai
,
G.
, and
Lin
,
Y.
,
1988
, “
A New Approximate Solution Technique for Randomly Excited Non-Linear Oscillators
,”
Int. J. Non-Linear Mech.
,
23
(
5–6
), pp.
409
–
420
.10.1016/0020-7462(88)90038-8
36.
Cai
,
G.
,
Lin
,
Y.
, and
Elishakoff
,
I.
,
1992
, “
A New Approximate Solution Technique for Randomly Excited Non-Linear Oscillatorsii
,”
Int. J. Non-Linear Mech.
,
27
(
6
), pp.
969
–
979
.10.1016/0020-7462(92)90049-D
37.
Di Paola
,
M.
, and
Pinnola
,
F. P.
,
2012
, “
Riesz Fractional Integrals and Complex Fractional Moments for the Probabilistic Characterization of Random Variables
,”
Probab. Eng. Mech.
,
29
, pp.
149
–
156
.10.1016/j.probengmech.2011.11.003
38.
Di Paola
,
M.
,
2014
, “
Fokker-Planck Equation Solved in Terms of Complex Fractional Moments
,”
Probab. Eng. Mech.
,
38
, pp.
70
–
76
.10.1016/j.probengmech.2014.09.003
39.
Jin
,
X.
,
Wang
,
Y.
,
Huang
,
Z.
, and
Di Paola
,
M.
,
2014
, “
Constructing Transient Response Probability Density of Non-Linear System Through Complex Fractional Moments
,”
Int. J. Non-Linear Mech.
,
65
, pp.
253
–
259
.10.1016/j.ijnonlinmec.2014.06.004
40.
Iwankiewicz
,
R.
,
Nielsen
,
S.
, and
Thoft-Christensen
,
P.
,
1990
, “
Dynamic Response of Non-Linear Systems to Poisson-Distributed Pulse Trains: Markov Approach
,”
Struct. Saf.
,
8
(
1–4
), pp.
223
–
238
.10.1016/0167-4730(90)90042-N
41.
Iwankiewicz
,
R.
, and
Nielsen
,
S.
,
1992
, “
Dynamic Response of Non-Linear Systems to Poisson-Distributed Random Impulses
,”
J. Sound Vib.
,
156
(
3
), pp.
407
–
423
.10.1016/0022-460X(92)90736-H
42.
Chechkin
,
A.
,
Gonchar
,
V.
,
Klafter
,
J.
,
Metzler
,
R.
, and
Tanatarov
,
L.
,
2002
, “
Stationary State of Non-Linear Oscillator Driven by Lévy Noise
,”
Chem. Phys.
,
284
(
1–2
), pp.
233
–
251
.10.1016/S0301-0104(02)00551-7
43.
Gonchar
,
V.
,
Tanatarov
,
L.
, and
Chechkin
,
A.
,
2002
, “
Stationary Solutions of the Fractional Kinetic Equation With a Symmetric Power-Law Potential
,”
Theor. Math. Phys.
,
131
(
1
), pp.
582
–
594
.10.1023/A:1015118206234
44.
Alotta
,
G.
, and
Di Paola
,
M.
,
2015
, “
Probabilistic Characterization of Nonlinear Systems Under α-Stable White Noise Via Complex Fractional Moments
,”
Phys. A: Stat. Mech. Its Appl.
,
420
, pp.
265
–
276
.10.1016/j.physa.2014.10.091
45.
Li
,
J.
, and
Chen
,
J. B.
,
2003
, “
Probability Density Evolution Method for Analysis of Stochastic Structural Dynamic Response
,”
Acta Mech. Sin.
,
35
(
4
), pp.
437
–
442
.
46.
Li
,
J.
, and
Chen
,
J.
,
2004
, “
Probability Density Evolution Method for Dynamic Response Analysis of Structures With Uncertain Parameters
,”
Comput. Mech.
,
34
(
5
), pp.
400
–
409
.10.1007/s00466-004-0583-8
47.
Li
,
J.
, and
Chen
,
J. B.
,
2006a
, “
The Probability Density Evolution Method for Dynamic Response Analysis of Non-Linear Stochastic Structures
,”
Int. J. Numer. Methods Eng.
,
65
(
6
), pp.
882
–
903
.10.1002/nme.1479
48.
Li
,
J.
, and
Chen
,
J.
,
2009
,
Stochastic Dynamics of Structures
, Stochastic Modeling Series.
Wiley
,
Singapore
.
49.
Li
,
J.
,
Sang
,
N.
, and
Gao
,
C.
,
2016
, “
Completed Local Similarity Pattern for Color Image Recognition
,”
Neurocomputing
,
182
, pp.
111
–
117
.10.1016/j.neucom.2015.12.005
50.
Li
,
J.
,
2016
,
Stochastic Dynamic Systems
,
Tonji University Press
,
Tonji, China
.
51.
Chen
,
G.
, and
Yang
,
D.
,
2019
, “
Direct Probability Integral Method for Stochastic Response Analysis of Static and Dynamic Structural Systems
,”
Comput. Methods Appl. Mech. Eng.
,
357
, p.
112612
.10.1016/j.cma.2019.112612
52.
Ricciardi
,
G.
, and
Elishakoff
,
I.
,
2002
, “
A Novel Local Stochastic Linearization Method Via Two Extremum Entropy Principles
,”
Int. J. Non-Linear Mech.
,
37
(
4–5
), pp.
785
–
800
.10.1016/S0020-7462(01)00099-3
53.
Inverardi
,
P.
, and
Tagliani
,
A.
,
2003
, “
Maximum Entropy Density Estimation From Fractional Moments
,”
Commun. Stat.-Theory Methods
,
32
(
2
), pp.
327
–
345
.10.1081/STA-120018189
54.
Alibrandi
,
U.
, and
Ricciardi
,
G.
,
2008
, “
Efficient Evaluation of the PDF of a Random Variable Through the Kernel Density Maximum Entropy Approach
,”
Int. J. Numer. Methods Eng.
,
75
(
13
), pp.
1511
–
1548
.10.1002/nme.2300
55.
Zhang
,
X.
,
He
,
W.
,
Zhang
,
Y.
, and
Pandey
,
M.
,
2017
, “
An Effective Approach for Probabilistic Lifetime Modelling Based on the Principle of Maximum Entropy With Fractional Moments
,”
Appl. Math. Modell.
,
51
, pp.
626
–
642
.10.1016/j.apm.2017.07.036
56.
Dai
,
H.
,
Zhang
,
R.
, and
Zhang
,
H.
,
2019
, “
A New Fractional Moment Equation Method for the Response Prediction of Nonlinear Stochastic Systems
,”
Nonlinear Dyn.
,
97
(
4
), pp.
2219
–
2230
.10.1007/s11071-019-05119-x
57.
Di Paola
,
M.
, and
Sofi
,
A.
,
2002
, “
Approximate Solution of the Fokker-Planck-Kolmogorov Equation
,”
Probab. Eng. Mech.
,
17
(
4
), pp.
369
–
384
.10.1016/S0266-8920(02)00034-6
58.
Di Paola
,
M.
, and
Failla
,
G.
,
2005
, “
Stochastic Response of Linear and Non-Linear Systems to α-Stable Lévy White Noises
,”
Probab. Eng. Mech.
,
20
(
2
), pp.
128
–
135
.10.1016/j.probengmech.2004.12.001
59.
Lacquaniti
,
S.
, and
Ricciardi
,
G.
,
2006
, “
A Probabilistic Linearization Method for Non-Linear Systems Subjected to Additive and Multiplicative Excitations
,”
Int. J. Non-Linear Mech.
,
41
(
10
), pp.
1191
–
1205
.10.1016/j.ijnonlinmec.2006.12.002
60.
Narayanan
,
S.
, and
Kumar
,
P.
,
2012
, “
Numerical Solutions of Fokker-Planck Equation of Nonlinear Systems Subjected to Random and Harmonic Excitations
,”
Probab. Eng. Mech.
,
27
(
1
), pp.
35
–
46
.10.1016/j.probengmech.2011.05.006
61.
Kapitaniak
,
T.
,
1985
, “
Stochastic Response With Bifurcations to Non-Linear Duffing's Oscillator
,”
J. Sound Vib.
,
102
(
3
), pp.
440
–
441
.10.1016/S0022-460X(85)80154-1
62.
Grigoriu
,
M.
,
1990
, “
Applications of Diffusion Models to Reliability Analysis of Daniels Systems
,”
Struct. Saf.
,
7
(
2–4
), pp.
219
–
228
.10.1016/0167-4730(90)90071-V
63.
Buss
,
A.
, and
Grigoriu
,
M.
,
1990
, “
Reliability of Linear Oscillators Subject to Wind Loads
,”
J. Wind Eng. Ind. Aerodyn.
,
36
(
1
), pp.
571
–
577
.10.1016/0167-6105(90)90339-E
64.
Naess
,
A.
, and
Johnsen
,
J. M.
,
1993
, “
Response Statistics of Nonlinear Compliant Offshore Structures by the Path Integral Solution Method
,”
Probab. Eng. Mech.
,
8
(
2
), pp.
91
–
106
.10.1016/0266-8920(93)90003-E
65.
Bergman
,
L.
,
Shinozuka
,
M.
,
Bucher
,
C.
,
Sobczyk
,
K.
,
Dasgupta
,
G.
,
Spanos
,
P.
,
Deodatis
,
G.
,
Spencer
,
B.
, Jr. ,
Ghanem
,
R.
,
Sutoh
,
A.
,
Grigoriu
,
M.
,
Takada
,
T.
,
Hoshiya
,
M.
,
Wedig
,
W.
,
Johnson
,
E.
,
Wojtkiewicz
,
S.
,
Naess
,
A.
,
Yoshida
,
I.
,
Pradlwarter
,
H.
,
Zeldin
,
B.
,
Schuller
,
G.
, and
Zhang
,
R.
,
1997
, “
A State-of-the-Art Report on Computational Stochastic Mechanics
,”
Probab. Eng. Mech.
,
12
(
4
), pp.
197
–
321
.
66.
Naess
,
A.
, and
Moe
,
V.
,
2000
, “
Efficient Path Integration Methods for Nonlinear Dynamic Systems
,”
Probab. Eng. Mech.
,
15
(
2
), pp.
221
–
231
.10.1016/S0266-8920(99)00031-4
67.
Iourtchenko
,
D.
,
Mo
,
E.
, and
Naess
,
A.
,
2008
, “
Reliability of Strongly Nonlinear Single Degree of Freedom Dynamic Systems by the Path Integration Method
,”
ASME J. Appl. Mech.
,
75
(
6
), p.
061016
.10.1115/1.2967896
68.
Iourtchenko
,
D. V.
,
Mo
,
E.
, and
Naess
,
A.
,
2006
, “
Response Probability Density Functions of Strongly Non-Linear Systems by the Path Integration Method
,”
Int. J. Non-Linear Mech.
,
41
(
5
), pp.
693
–
705
.10.1016/j.ijnonlinmec.2006.04.002
69.
Barone
,
G.
,
Navarra
,
G.
, and
Pirrotta
,
A.
,
2008
, “
Probabilistic Response of Linear Structures Equipped With Nonlinear Damper Devices (PIS Method)
,”.
Probab. Eng. Mech.
,
23
(
2–3
), pp.
125
–
133
.10.1016/j.probengmech.2007.12.025
70.
Naess
,
A.
,
Iourtchenko
,
D.
, and
Batsevych
,
O.
,
2011
, “
Reliability of Systems With Randomly Varying Parameters by the Path Integration Method
,”
Probab. Eng. Mech.
,
26
(
1
), pp.
5
–
9
.10.1016/j.probengmech.2010.05.005
71.
Alevras
,
P.
, and
Yurchenko
,
D. V.
,
2016
, “
GPU Computing for Accelerating the Numerical Path Integration Approach
,”
Comput. Struct.
,
171
, pp.
46
–
53
.10.1016/j.compstruc.2016.05.002
72.
Grigoriu
,
M.
,
1995
, “
Linear and Nonlinear Systems With Non-Gaussian White Noise Input
,”
Probab. Eng. Mech.
,
10
(
3
), pp.
171
–
179
.10.1016/0266-8920(95)00014-P
73.
Di Paola
,
M.
, and
Santoro
,
R.
,
2008
, “
Non-Linear Systems Under Poisson White Noise Handled by Path Integral Solution
,”
J. Vib. Control
,
14
(
1–2
), pp.
35
–
49
.10.1177/1077546307079386
74.
Janakiraman
,
D.
, and
Sebastian
,
K. L.
,
2015
, “
Phase Space Path Integral Approach to Harmonic Oscillator With a Time-Dependent Force Constant
,”
Phys. A: Stat. Mech. Its Appl.
,
433
, pp.
198
–
203
.10.1016/j.physa.2015.01.047
75.
Di Matteo
,
A.
,
Di Paola
,
M.
, and
Pirrotta
,
A.
,
2016
, “
Path Integral Solution for Nonlinear Systems Under Parametric Poissonian White Noise Input
,”
Probab. Eng. Mech.
,
44
, pp.
89
–
98
.10.1016/j.probengmech.2015.09.020
76.
Zhu
,
H. T.
, and
Duan
,
L. L.
,
2016
, “
Probabilistic Solution of Non-Linear Random Ship Roll Motion by Path Integration
,”
Int. J. Non-Linear Mech.
,
83
, pp.
1
–
8
.10.1016/j.ijnonlinmec.2016.03.010
77.
Di Matteo
,
A.
, and
Pirrotta
,
A.
,
2017
, “
Path Integral Method for Nonlinear Systems Under Lévy White Noise
,”
ASME
Paper No. RISK-16-1095. 10.11115/RISK-16-1095
78.
Iwankiewicz
,
R.
, and
Nielsen
,
S. R. K.
,
1996
, “
Dynamic Response of Non-Linear Systems to Renewal Impulses by Path Integration
,”
J. Sound Vib.
,
195
(
2
), pp.
175
–
193
.10.1006/jsvi.1996.0415
79.
Iwankiewicz
,
R.
, and
Nielsen
,
S. R. K.
,
2000
, “
Solution Techniques for Pulse Problems in Nonlinear Stochastic Dynamics
,”
Probab. Eng. Mech.
,
15
(
1
), pp.
25
–
36
.10.1016/S0266-8920(99)00006-5
80.
Iwankiewicz
,
R.
, and
Di Paola
,
M.
,
2019
, “
Probability Density of Response of Dynamic Systems to Renewal Impulse Processes: Path Integral Solution
,”
Fourth Polish Congress of Mechanics—23rd International Conference on Computer Methods in Mechanic
, Krakow, Poland, Sept. 9, p.
020018
.
81.
Dimentberg
,
M.
,
Gaidai
,
O.
, and
Naess
,
A.
,
2009
, “
Random Vibrations With Strongly Inelastic Impacts: Response PDF by the Path Integration Method
,”
Int. J. Non-Linear Mech.
,
44
(
7
), pp.
791
–
796
.10.1016/j.ijnonlinmec.2009.04.007
82.
Cottone
,
G.
,
Di Paola
,
M.
,
Ibrahim
,
R.
,
Pirrotta
,
A.
, and
Santoro
,
R.
,
2009
, “
Ship Roll Motion Under Stochastic Agencies Using Path Integral Method
,”
Lecture Notes Appl. Comput. Mech.
,
44
, pp.
29
–
40
.10.1007/978-3-642-00629-6_4
83.
Cottone
,
G.
,
Di Paola
,
M.
,
Ibrahim
,
R.
,
Pirrotta
,
A.
, and
Santoro
,
R.
,
2010
, “
Stochastic Ship Roll Motion Via Path Integral Method
,”
Int. J. Nav. Architecture Ocean Eng.
,
2
(
3
), pp.
119
–
126
.10.2478/IJNAOE-2013-0027
84.
Kougioumtzoglou
,
I. A.
, and
Spanos
,
P. D.
,
2013
, “
Response and First-Passage Statistics of Nonlinear Oscillators Via a Numerical Path Integral Approach
,”
J. Eng. Mech.
,
139
(
9
), pp.
1207
–
1217
.10.1061/(ASCE)EM.1943-7889.0000564
85.
Kougioumtzoglou
,
I.
, and
Spanos
,
P.
,
2014
, “
Stochastic Response Analysis of the Softening Duffing Oscillator and Ship Capsizing Probability Determination Via a Numerical Path Integral Approach
,”
Probab. Eng. Mech.
,
35
, pp.
67
–
74
.10.1016/j.probengmech.2013.06.001
86.
Di Paola
,
M.
, and
Bucher
,
C.
,
2016
, “
Ideal and Physical Barrier Problems for Non-Linear Systems Driven by Normal and Poissonian White Noise Via Path Integral Method
,”
Int. J. Non-Linear Mech.
,
81
, pp.
274
–
282
.10.1016/j.ijnonlinmec.2016.01.008
87.
Bucher
,
C.
,
Di Matteo
,
A.
,
Di Paola
,
M.
, and
Pirrotta
,
A.
,
2016
, “
First-Passage Problem for Nonlinear Systems Under Lévy White Noise Through Path Integral Method
,”
Nonlinear Dyn.
,
85
(
3
), pp.
1445
–
1456
.10.1007/s11071-016-2770-9
88.
Li
,
J.
, and
Chen
,
J. B.
,
2006b
, “
The Dimension-Reduction Strategy Viamapping for the Probability Density Evolution Analysis of Nonlinear Stochastic Systems
,”
Probab. Eng. Mech.
,
21
(
4
), pp.
442
–
453
.10.1016/j.probengmech.2006.02.004
89.
Wiener
,
N.
,
1921
, “
The Average of an Analytic Functional
,”
Proc. Natl. Aced. Sci.
,
7
(
9
), pp.
253
–
260
.10.1073/pnas.7.9.253
90.
Taniguchi
,
T.
, and
Cohen
,
E. G. D.
,
2007
, “
Inertial Effects in Nonequilibrium Work Fluctuations by a Path Integral Approach
,”
J. Stat. Phys.
,
130
(
1
), pp.
1
–
26
.10.1007/s10955-007-9398-6
91.
Chaichian
,
M.
, and
Demichev
,
A.
,
2001
,
Path Integrals in Physics, Vol. I Stochastic Processes and Quantum Mechanics
,
Institute of Physics Publishing
,
Bristol, UK; Philadelphia, PA
.
92.
Feynman
,
R. P.
,
1948
, “
Space-Time Approach to Non-Relativistic Quantum Mechanics
,”
Rev. Mod. Phys.
,
20
(
2
), pp.
367
–
387
.10.1103/RevModPhys.20.367
93.
Wehner
,
M. F.
, and
Wolfer
,
W. G.
,
1983
, “
Numerical Evaluation of Path-Integral Solutions to Fokker-Planck Equations—II: Restricted Stochastic Processes
,”
Phys. Rev. A
,
28
(
5
), pp.
3003
–
3011
.10.1103/PhysRevA.28.3003
94.
Kougioumtzoglou
,
I. A.
, and
Spanos
,
P. D.
,
2012
, “
An Analytical Wiener Path Integral Technique for Non-Stationary Response Determination of Nonlinear Oscillators
,”
Probab. Eng. Mech.
,
28
, pp.
125
–
131
.10.1016/j.probengmech.2011.08.022
95.
Kougioumtzoglou
,
I. A.
, and
Spanos
,
P. D.
,
2014
, “
Non-Stationary Stochastic Response Determination of Nonlinear Systems: A Wiener Path Integral Formalism
,”
ASCE J. Eng. Mech.
,
140
(
9
), p.
04014064
.10.1061/(ASCE)EM.1943-7889.0000780
96.
Kougioumtzoglou
,
I. A.
,
Di Matteo
,
A.
,
Spanos
,
P. D.
,
Pirrotta
,
A.
, and
Di Paola
,
M.
,
2015
, “
An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems
,”
ASME J. Appl. Mech.
,
82
(
10
), p.
101005
.10.1115/1.4030890
97.
Kougioumtzoglou
,
I. A.
,
2017
, “
A Wiener Path Integral Solution Treatment and Effective Material Properties of a Class of One-Dimensional Stochastic Mechanics Problems
,”
ASCE J. Eng. Mech.
,
143
(
6
), p.
04017014
.10.1061/(ASCE)EM.1943-7889.0001211
98.
Psaros
,
A. F.
,
Kougioumtzoglou
,
I. A.
, and
Petromichelakis
,
I.
,
2018
, “
Sparse Representations and Compressive Sampling for Enhancing the Computational Efficiency of the Wiener Path Integral Technique
,”
Mech. Syst. Signal Process.
,
111
, pp.
87
–
101
.10.1016/j.ymssp.2018.03.056
99.
Petromichelakis
,
I.
,
Psaros
,
A. F.
, and
Kougioumtzoglou
,
I. A.
,
2018
, “
Stochastic Response Determination and Optimization of a Class of Nonlinear Electromechanical Energy Harvesters: A Wiener Path Integral Approach
,”
Probab. Eng. Mech.
,
53
, pp.
116
–
125
.10.1016/j.probengmech.2018.06.004
100.
Psaros
,
A. F.
,
Petromichelakis
,
I.
, and
Kougioumtzoglou
,
I. A.
,
2019
, “
Wiener Path Integrals and Multi-Dimensional Global Bases for Non-Stationary Stochastic Response Determination of Structural Systems
,”
Mech. Syst. Signal Process.
,
128
, pp.
551
–
571
.10.1016/j.ymssp.2019.04.014
101.
Psaros
,
A. F.
,
Brudastova
,
O.
,
Malara
,
G.
, and
Kougioumtzoglou
,
I. A.
,
2018
, “
Wiener Path Integral Based Response Determination of Nonlinear Systems Subject to Non-White, non-Gaussian, and Non-Stationary Stochastic Excitation
,”
J. Sound Vib.
,
433
, pp.
314
–
333
.10.1016/j.jsv.2018.07.013
102.
Meimaris
,
A. T.
,
Kougioumtzoglou
,
I. A.
, and
Pantelous
,
A. A.
,
2018
, “
A Closed Form Approximation and Error Quantification for the Response Transition Probability Density Function of a Class of Stochastic Differential Equations
,”
Probab. Eng. Mech.
,
54
, pp.
87
–
94
.10.1016/j.probengmech.2017.07.005
103.
Meimaris
,
A. T.
,
Kougioumtzoglou
,
I. A.
, and
Pantelous
,
A. A.
,
2019
, “
Approximate Analytical Solutions for a Class of Nonlinear Stochastic Differential Equations
,”
Eur. J. Appl. Math.
,
30
(
5
), pp.
928
–
944
.10.1017/S0956792518000530
104.
Meimaris
,
A. T.
,
Kougioumtzoglou
,
I. A.
,
Pantelous
,
A. A.
, and
Pirrotta
,
A.
,
2019
, “
An Approximate Technique for Determining in Closed Form the Response Transition Probability Density Function of Diverse Nonlinear/Hysteretic Oscillators
,”
Nonlinear Dyn.
,
97
(
4
), pp.
2627
–
2641
.10.1007/s11071-019-05152-w
105.
Petromichelakis
,
I.
,
Psaros
,
A. F.
, and
Kougioumtzoglou
,
I. A.
,
2020
, “
Stochastic Response Determination of Nonlinear Structural Systems With Singular Diffusion Matrices: A Wiener Path Integral Variational Formulation With Constraints
,”
Probab. Eng. Mech.
,
60
, p.
103044
.10.1016/j.probengmech.2020.103044
106.
Psaros
,
A. F.
,
Zhao
,
Y.
, and
Kougioumtzoglou
,
I. A.
,
2020
, “
An Exact Closed-Form Solution for Linear Multi-Degree-of-Freedom Systems Under Gaussian White Noise Via the Wiener Path Integral Technique
,”
Probab. Eng. Mech.
,
60
, p.
103040
.10.1016/j.probengmech.2020.103040
107.
Di Matteo
,
A.
,
Kougioumtzoglou
,
I. A.
,
Pirrotta
,
A.
,
Spanos
,
P. D.
, and
Di Paola
,
M.
,
2014
, “
Stochastic Response Determination of Nonlinear Oscillators With Fractional Derivatives Elements Via the Wiener Path Integral
,”
Probab. Eng. Mech.
,
38
, pp.
127
–
135
.10.1016/j.probengmech.2014.07.001
108.
Almeida
,
R.
, and
Torres
,
D. F.
,
2011
, “
Necessary and Sufficient Conditions for the Fractional Calculus of Variations With Caputo Derivatives
,”
Commun. Nonlinear Sci. Numer. Simul.
,
16
(
3
), pp.
1490
–
1500
.10.1016/j.cnsns.2010.07.016
109.
Agrawal
,
O. P.
,
2007
, “
Fractional Variational Calculus in Terms of the Riesz Fractional Derivatives
,”
J. Phys. A: Math. Theoretica
,
40
(
24
), pp.
6287
–
6303
.10.1088/1751-8113/40/24/003
110.
Tarasov
,
V.
,
2006
, “
Fractional Variations for Dynamical Systems: Hamilton and Lagrange Approaches
,”
J. Phys. A: Math. General
,
39
(
26
), pp.
8409
–
8425
.10.1088/0305-4470/39/26/009
111.
Di Paola
,
M.
, and
Ricciardi
,
G.
,
1992
, “
Vibration of a Bridge Under a Random Train of Moving Loads
,”
Proceedings of the Specialty Conference on Probabilistic Mechanics and Structural and Geotechnical Reliability
, Denver, CO, pp.
136
–
139
.https://cedb.asce.org/CEDBsearch/record.jsp?dockey=0077369
112.
Ricciardi
,
G.
,
1994
, “
Random Vibration of Beam Under Moving Loads
,”
J. Eng. Mech.
,
120
(
11
), pp.
2361
–
2379
.10.1061/(ASCE)0733-9399(1994)120:11(2361)
113.
Muscolino
,
G.
, and
Ricciardi
,
G.
,
1996
, “
Probability Density Function of Linear Systems Subjected to a Random Stream of Poisson Pulses
,”
Proceedings of the Specialty Conference on Probabilistic Mechanics and Structural and Geotechnical Reliability
, pp.
388
–
391
.
114.
Muscolino
,
G.
, and
Ricciardi
,
G.
,
1999
, “
Probability Density Function of MDOF Structural Systems Under Non-Normal Delta-Correlated Inputs
,”
Comput. Methods Appl. Mech. Eng.
,
168
(
1–4
), pp.
121
–
133
.10.1016/S0045-7825(98)00137-6
115.
Muscolino
,
G.
,
Ricciardi
,
G.
, and
Cacciola
,
P.
,
2003
, “
Monte Carlo Simulation in the Stochastic Analysis of Non-Linear Systems Under External Stationary Poisson White Noise Input
,”
Int. J. Non-Linear Mech.
,
38
(
8
), pp.
1269
–
1283
.10.1016/S0020-7462(02)00072-0
116.
Grigoriu
,
M.
,
2004
, “
Characteristic Function Equation for the State of Dynamic System With Gaussian, Poisson and Lévy White Noise
,”
Probababilistic Eng. Mech.
,
19
(
4
), pp.
449
–
461
.10.1016/j.probengmech.2004.05.003
117.
Jespersen
,
S.
,
Metzler
,
R.
, and
Fogedby
,
H. C.
,
1999
, “
Lévy Flights in External Force Fields: Langevin and Fractional Fokker–Planck Equations and Their Solutions
,”
Phys. Rev. E
,
59
(
3
), pp.
2736
–
2745
.10.1103/PhysRevE.59.2736
118.
Vasta
,
M.
,
1995
, “
Exact Stationary Solution for a Class of Non-Linear Systems Driven by a Non-Normal Delta-Correlated Process
,”
Int. J. Non-Linear Mech.
,
30
(
4
), pp.
407
–
418
.10.1016/0020-7462(95)00009-D
119.
Proppe
,
C.
,
2003
, “
Exact Stationary Probability Density Functions for Non-Linear Systems Under Poisson White Noise Excitation
,”
Int. J. Non-Linear Mech.
,
38
(
4
), pp.
557
–
564
.10.1016/S0020-7462(01)00084-1
120.
Caddemi
,
S.
, and
Di Paola
,
M.
,
1996
, “
Ideal and Physical White Noise in Stochastic Analysis
,”
Int. J. Non-Linear Mech.
,
31
(
5
), pp.
581
–
590
.10.1016/0020-7462(96)00023-6
121.
Caddemi
,
S.
, and
Di Paola
,
M.
,
1997
, “
Nonlinear System Response for Impulsive Parametric Input
,”
ASME J. Appl. Mech.
,
64
(
3
), pp.
642
–
648
.10.1115/1.2788941
122.
Pirrotta
,
A.
,
2005
, “
Non-Linear Systems Under Parametric White Noise Input: Digital Simulation and Response
,”
Int. J. Non-Linear Mech.
,
40
(
8
), pp.
1088
–
1101
.10.1016/j.ijnonlinmec.2005.04.001
123.
Gardiner
,
C.
,
1994
,
Handbook of Stochastic Methods
, 4th ed.,
Springer-Verlag
,
Berlin
.
124.
Yu
,
J.
,
Cai
,
G. Q.
, and
Lin
,
Y. K.
,
1997
, “
A New Path Integration Procedure Based on Gauss-Legendre Scheme
,”
Int. J. Nonlinear Mech.
,
32
(
4
), pp.
759
–
768
.10.1016/S0020-7462(96)00096-0
125.
Ren
,
Z.
, and
Xu
,
W.
,
2020
, “
An Improved Path Integration Method for Nonlinear Systems Under Poisson White Noise Excitation
,”
Appl. Math. Comput.
,
373
, p.
125036
.10.1016/j.amc.2020.125036
126.
Ren
,
Z.
,
Xu
,
W.
, and
Zhang
,
S.
,
2020
, “
Reliability Analysis of Nonlinear Vibro-Impact Systems With Both Randomly Fluctuating Restoring and Damping Terms
,”
Commun. Nonlinear Sci. Numer. Simul.
,
82
, p.
105087
.10.1016/j.cnsns.2019.105087
127.
Naess
,
A.
,
Kolnes
,
F.
, and
Mo
,
E.
,
2007
, “
Stochastic Spur Gear Dynamics by Numerical Path Integration
,”
J. Sound Vib.
,
302
(
4–5
), pp.
936
–
950
.10.1016/j.jsv.2006.12.017
128.
Di Paola
,
M.
, and
Santoro
,
R.
,
2009
, “
Path Integral Solution Handled by Fast Gauss Transform
,”
Probab. Eng. Mech.
,
24
(
3
), pp.
300
–
311
.10.1016/j.probengmech.2008.07.008
129.
Di Matteo
,
A.
,
2019
, “
Path Integral Approach Via Laplace's Method of Integration for Nonstationary Response of Nonlinear Systems
,”
Meccanica
,
54
(
9
), pp.
1351
–
1363
.10.1007/s11012-019-00991-8
130.
Di Paola
,
M.
,
Pirrotta
,
A.
, and
Zingales
,
M.
,
2007
, “
Itô Calculus Extended to Systems Driven by α-Stable Lévy White Noises (a Novel Clip on the Tails of Lévy Motion)
,”.
Int. J. Non-Linear Mech.
,
42
(
8
), pp.
1046
–
1054
.10.1016/j.ijnonlinmec.2007.07.001
131.
Naess
,
A.
, and
Skaug
,
C.
,
2001
, “
Path Integration Methods for Calculating Response Statistics of Nonlinear Oscillators Driven by α-Stable Lévy Noise
,” IUTAM Symposium on Nonlinear and Stochastic Structural Dynamics,
S.
Narayanan
, and
R.
Iyengar
, eds.,
Kluwer Academic Publishers,
Madras, India, Jan. 4-8, pp.
159
–
169
.
132.
Xu
,
Y.
,
Zan
,
W.
,
Jia
,
W.
, and
Kurths
,
J.
,
2019
, “
Path Integral Solutions of the Governing Equation of SDEs Excited by Lévy White Noise
,”
J. Comput. Phys.
,
394
, pp.
41
–
55
.10.1016/j.jcp.2019.05.023
133.
Grigoriu
,
M.
,
1995
,
Applied Non-Gaussian Processes: Examples, Theory, Simulation, Linear Random Vibration, and MATLAB Solutions
,
Prentice Hall
,
Wellington, New Zealand
.
134.
Falsone
,
G.
, and
Settineri
,
D.
,
2013
, “
Explicit Solutions for the Response Probability Density Function of Linear Systems Subjected to Random Static Loads
,”
Probab. Eng. Mech.
,
33
, pp.
86
–
94
.10.1016/j.probengmech.2013.03.001
135.
Settineri
,
D.
, and
Falsone
,
G.
,
2014
, “
A Method for the Evaluation of the Response Probability Density Function of Some Linear Dynamic Systems Subjected to non-Gaussian Random Load
,”
Probab. Eng. Mech.
,
38
, pp.
165
–
172
.10.1016/j.probengmech.2014.03.008
136.
Di Paola
,
M.
, and
Falsone
,
G.
,
1993
, “
Stochastic Dynamics of Nonlinear Systems Driven by Non-Normal Delta-Correlated Processes
,”
ASME J. Appl. Mech.
,
60
(
1
), pp.
141
–
148
.10.1115/1.2900736
137.
Di Paola
,
M.
, and
Falsone
,
G.
,
1993
, “
Itô and Stratonovich Integrals for Delta-Correlated Processes
,”
Probab. Eng. Mech.
,
8
(
3–4
), pp.
197
–
208
.10.1016/0266-8920(93)90015-N
138.
Di Paola
,
M.
, and
Pirrotta
,
A.
,
1999
, “
Non-Linear Systems Under Impulsive Parametric Input
,”
Int. J. Non-Linear Mech.
,
34
(
5
), pp.
843
–
851
.10.1016/S0020-7462(98)00057-2
139.
Pirrotta
,
A.
,
2007
, “
Multiplicative Cases From Additive Cases: Extension of Kolmogorov-Feller Equation to Parametric Poisson White Noise Processes
,”
Probab. Eng. Mech.
,
22
(
2
), pp.
127
–
135
.10.1016/j.probengmech.2006.08.005
140.
Di Matteo
,
A.
,
Di Paola
,
M.
, and
Pirrotta
,
A.
,
2017
, “
Direct Evaluation of Jumps for Nonlinear Systems Under External and Multiplicative Impulses
,”
J. Vib. Control
,
23
(
11
), pp.
1753
–
1767
.10.1177/1077546315600111
141.
Köylüoğlu
,
H. U.
,
Nielsen
,
S. R. K.
, and
Iwankiewicz
,
R.
,
1995
, “
Response and Reliability of Poisson-Driven Systems by Path Integration
,”
J. Eng. Mech.
,
121
(
1
), pp.
117
–
130
.10.1061/(ASCE)0733-9399(1995)121:1(117)
142.
Crandall
,
S.
,
1970
, “
First-Crossing Probabilities of the Linear Oscillator
,”
J. Sound Vib.
,
12
(
3
), pp.
285
–
299
.10.1016/0022-460X(70)90073-8
143.
Darling
,
D. A.
, and
Siegert
,
A. J. F.
,
1953
, “
The First Passage Problem for a Continuous Markov Process
,”
Ann. Math. Statist.
,
24
(
4
), pp.
624
–
639
.10.1214/aoms/1177728918
144.
Gray
,
A. H.
,
1966
, “
First-Passage Time in a Random Vibrational System
,”
ASME J. Appl Mech
,
33
(
1
), pp.
187
–
191
.10.1115/1.3624977
145.
Roberts
,
J.
,
1976
, “
First Passage Time for the Envelope of a Randomly Excited Linear Oscillator
,”
J. Sound Vib.
,
46
(
1
), pp.
1
–
14
.10.1016/0022-460X(76)90812-9
146.
Siegert
,
A. J. F.
,
1951
, “
On the First Passage Time Probability Problem
,”
Phys. Rev.
,
81
(
4
), pp.
617
–
623
.10.1103/PhysRev.81.617
147.
Rice
,
J. R.
, and
Beer
,
F. P.
,
1966
, “
First-Occurrence Time of High-Level Crossings in a Continuous Random Process
,”
J. Acoust. Soc. Am.
,
39
(
2
), pp.
323
–
335
.10.1121/1.1909893
148.
Köylüoglu
,
H. U.
,
Nielsen
,
S. R. K.
, and
Iwankiewicz
,
R.
,
1994
, “
Reliability of Non-Linear Oscillators Subject to Poisson Driven Impulses
,”
J. Sound Vib.
,
176
(
1
), pp.
19
–
33
.10.1006/jsvi.1994.1356
149.
Holstein
,
B. R.
, and
Swift
,
A. R.
,
1982
, “
Barrier Penetration Via Path Integrals
,”
Am. J. Phys.
,
50
(
9
), pp.
833
–
839
.10.1119/1.12751
150.
Li
,
C.
,
Xu
,
W.
, and
Yue
,
X.
,
2014
, “
Stochastic Response of a Vibro-Impact System by Path Integration Based on Generalized Cell Mapping Method
,”
Int. J. Bifurcation Chaos
,
24
(
10
), p.
1450129
.10.1142/S0218127414501296
151.
Gikhman
,
I. I.
, and
Skorohod
,
A. V.
,
2004
,
The Theory of Stochastic Processes
, Vol.
II
,
Springer-Verlag
,
Berlin
.
152.
Wiener
,
N.
,
1923
, “
Differential-Space
,”
J. Math. Phys.
,
2
(
1–4
), pp.
131
–
174
.10.1002/sapm192321131
153.
Wiener
,
N.
,
1924
, “
The Dirichlet Problem
,”
J. Math. Phys.
,
3
(
3
), pp.
127
–
146
.10.1002/sapm192433127
154.
Wiener
,
N.
, and
Paley
,
R. C.
,
1934
,
Fourier Transforms in the Complex Domain
,
Colloquium Publications, American Mathematical Society
,
Providence, RI
.
155.
Kougioumtzoglou
,
I.
, and
Spanos
,
P.
,
2009
, “
An Approximate Approach for Nonlinear System Response Determination Under Evolutionary Stochastic Excitation
,”
Curr. Sci.
,
97
(
8
), pp.
1203
–
1211
.
156.
Ewing
,
G. M.
,
1985
,
Calculus of Variations With Applications
,
Dover
,
New York
.
157.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
Academic Press
,
San Diego, CA
.
158.
Samko
,
S.
,
Kilbas
,
A.
, and
Marichev
,
O.
,
1993
,
Fractional Integrals and Derivatives
,
Gordon&Breach Science Publisher
,
Amsterdam, The Netherland
.
159.
Colinas-Armijo
,
N.
, and
Di Paola
,
M.
,
2018
, “
Step-by-Step Integration for Fractional Operators
,”
Commun. Nonlinear Sci. Numer. Simul.
,
59
, pp.
292
–
305
.10.1016/j.cnsns.2017.11.030
160.
Baleanu
,
D.
,
Diethelm
,
K.
,
Scalas
,
E.
, and
Trujillo
,
J. J.
,
2016
,
A Survey of Numerical Methods for the Solution of Ordinary and Partial Fractional Differential Equations
, World Scientific, Singapore, pp.
39
–
94
.
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