Document Zbl 07901049

Time fractional equations and anomalous sub-diffusions – in memory of Professor Shisong Mao. (English) Zbl 07901049

Chin. J. Appl. Probab. Stat. 40, No. 2, 323-342 (2024).

Summary: In this paper, we survey some recent progress in the study of time fractional equations and its interplay with anomalous sub-diffusions, with some improvements and extensions.

MSC:

26A33	Fractional derivatives and integrals
60H30	Applications of stochastic analysis (to PDEs, etc.)
34K37	Functional-differential equations with fractional derivatives
01A70	Biographies, obituaries, personalia, bibliographies

Keywords:

time fractional derivative; time fractional equation; subordinator; inverse subordinator; strong and weak solution

Biographic References:

Mao, Shisong

Cite Review PDF

Full Text: DOI

References:

[1]	MEERSCHAERT M M, SCHEFFLER H P. Limit theorems for continuous-time random walks with infinite mean waiting times [J]. J Appl Probab, 2004, 41(3): 623-638. · Zbl 1065.60042
[2]	KLAFTER J, SOKOLOV I M. Anomalous diffusion spreads its wings [J]. Phys World, 2005, 18(8): 29-32.
[3]	METZLER R, KLAFTER J. The random walk’s guide to anomalous diffusion: a fractional dynamics approach [J]. Phys Pep, 2000, 339(1): 1-77. · Zbl 0984.82032
[4]	PIRYATINSKA A, SAICHEV A I, WOYCZYNSKI W A. Models of anomalous diffusion: the subd-iffusive case [J]. Phys A, 2005, 349(3-4): 375-420.
[5]	HERRMANN R. Fractional Calculus: An Introduction for Physicists [M]. 2nd ed. Singapore: World Scientific Publishing, 2014. · Zbl 1293.26001
[6]	METZLER R, KLAFTER J. The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics [J]. J Pyhs A, 2004, 37(31): R161-R208. · Zbl 1075.82018
[7]	MEERSCHAERT M M, SIKORSKII A. Stochastic Models for Fractional Calculus [M]. Berlin: De Gruyter, 2012. · Zbl 1247.60003
[8]	SHLESINGER M F, KLAFTER J, WONG Y M. Random walks with infinite spatial and temporal moments [J]. J Statist Phys, 1982, 27(3): 499-512. · Zbl 0521.60080
[9]	BAEUMER B, MEERSCHAERT M M. Stochastic solutions for fractional Cauchy problems [J]. Fract Calc Appl Anal, 2001, 4(4): 481-500. · Zbl 1057.35102
[10]	HERN ÁNDEZ-HERN ÁNDEZ M E, KOLOKOLTSOV V N, TONIAZZI L. Generalised fractional evolution equations of Caputo type [J]. Chaos Solitons Fractals, 2017, 102: 184-196. · Zbl 1374.34009
[11]	KOLOKOLTSOV V N. On fully mixed and multidimensional extensions of the Caputo and Riemann-Liouville derivatives, related Markov processes and fractional differential equations [J]. Fract Calc Appl Anal, 2015, 18(4): 1039-1073. · Zbl 1321.26013
[12]	KOLOKOLTSOV V N. The probabilistic point of view on the generalized fractional partial differential equations [J]. Fract Calc Appl Anal, 2019, 22(3): 543-600. · Zbl 1483.35002
[13]	CHEN Z-Q. Time fractional equations and probabilistic representation [J]. Chaos Solitons Fractals, 2017, 102: 168-174. arXiv version with notes added after publication: arXiv:1703.01739v2 [math.PR]. · Zbl 1374.60122
[14]	No. 2 CHEN Z.-Q.: Time Fractional Equations and Anomalous Sub-Diffusions 341
[15]	MILLER K S, ROSS B. An Introduction to the Fractional Calculus and Fractional Differential Equa-tions [M]. New York: Wiley, 1993. · Zbl 0789.26002
[16]	SAMKO S G, KILBAS A A, MARICHEV O I. Fractional Integrals and Derivatives: Theory and Applications [M]. Switzerland: Gordon and Breach Science Publishers, 1993. · Zbl 0818.26003
[17]	UCHAIKIN V V. Fractional Derivatives for Physicists and Engineers: Volume I Background and Theory [M]. Berlin: Springer and Beijing: Higher Education Press, 2013. · Zbl 1312.26002
[18]	UCHAIKIN V V. Fractional Derivatives for Physicists and Engineers: Volume II Applications [M]. Berlin: Springer and Beijing: Higher Education Press, 2013. · Zbl 1312.26002
[19]	CHEN Z-Q, KIM P, KUMAGAI T, et al. Heat kernel estimates for time fractional equations [J]. Forum Math, 2018, 30(5): 1163-1192. · Zbl 1401.60088
[20]	CHEN Z-Q, KIM P, KUMAGAI T, et al. Time fractional Poisson equations: representations and estimates [J]. J Funct Anal, 2020, 278(2): 108311 (48 pages). · Zbl 1427.35312
[21]	BERTOIN J, YOR M. Increasing processes and the change of variables formula for non-decreasing functions [OL]. 2013 [2013-3-26]. https://hal.science/hal-00804781.
[22]	CHEN Z-Q. Gaugeability and conditional gaugeability [J]. Trans Amer Math Soc, 2002, 354(11): 4639-4679. · Zbl 1006.60072
[23]	CHEN Z-Q, FITZSIMMONS P J, KUWAE K, et al. Perturbation of symmetric Markov processes [J]. Probab Theory Related Fields, 2008, 140(1-2): 239-275. · Zbl 1137.31004
[24]	CHEN Z-Q, SONG R M. Conditional gauge theorem for non-local Feynman-Kac transforms [J]. Proba-b Theory Related Fields, 2003, 125(1): 45-72. · Zbl 1025.60032
[25]	TOALDO B. Convolution-type derivatives, hitting-times of subordinators and time-changed C0-semigroups [J]. Potential Anal, 2015, 42(1): 115-140. · Zbl 1315.60050
[26]	EIDELMAN S D, KOCHUBEI A N. Cauchy problem for fractional diffusion equations [J]. J Differ-ential Equations, 2004, 199(2): 211-255. · Zbl 1068.35037
[27]	TONIAZZI L. Stochastic classical solutions for space-time fractional evolution equations on a bounded domain [J]. J Math Anal Appl, 2019, 469(2): 594-622. · Zbl 1401.60126
[28]	BERTOIN J. Lévy Processes [M].
[29]	CHEN Z-Q, KIM K H, KIM P. Fractional time stochastic partial differential equations [J]. Stochastic Process Appl, 2015, 125(4): 1470-1499. · Zbl 1322.60106
[30]	MIJENA J B, NANE E. Space-time fractional stochastic partial differential equations [J]. Stochastic Process Appl, 2015, 125(9): 3301-3326. · Zbl 1329.60216
[31]	LIU W, R ÖCKNER M, DA SILVA J L. Quasi-linear (stochastic) partial differential equations with time-fractional derivatives [J]. SIAM J Math Anal, 2018, 50(3): 2588-2607. · Zbl 1407.60086
[32]	ASOGWA S A, MIJENA J B, NANE E. Blow-up results for space-time fractional stochastic partial differential equations [J]. Potential Anal, 2020, 53(2): 357-386. · Zbl 1453.60113
[33]	CHEN Z-Q, DENG W H, XU P B. Feynman-Kac transform for anomalous processes [J]. SIAM J Math Anal, 2021, 53(5): 6017-6047. · Zbl 1481.60203
[34]	ZHANG S Q, CHEN Z-Q. Fokker-Planck equation for Feynman-Kac transform of anomalous pro-cesses [J]. Stochastic Process Appl, 2022, 147: 300-326. · Zbl 1483.60151
[35]	KOBAYASHI K. Stochastic calculus for a time-changed semimartingale and the associated stochastic differential equations [J]. J Theoret Probab, 2011, 24(3): 789-820. · Zbl 1252.60054
[36]	NANE E, NI Y N. Path stability of stochastic differential equations driven by time-changed Lévy noises [J]. ALEA Lat Am J Probab Math Stat, 2018, 15(1): 479-507. · Zbl 06866535
[37]	LI Z, XU L P, YAN L T. McKean-Vlasov stochastic differential equations driven by the time-changed Brownian motion [J]. J Math Anal Appl, 2023, 527(1): 127336 (20 pages). · Zbl 1522.60052
[38]	ZHANG S Q, CHEN Z-Q. Fully coupled forward-backward stochastic differential equations driven by sub-diffusions [OL]. 2023 [2023-11-26]. https://arxiv.org/abs/2311.15151.
[39]	KAMOCKI R, MAJEWSKI M. Fractional linear control systems with Caputo derivative and their optimization [J]. Optimal Control Appl Methods, 2015, 36(6): 953-967. · Zbl 1333.93124
[40]	NANE E, NI Y N. A time-changed stochastic control problem and its maximum principle [J]. Probab Math Statist, 2021, 41(2): 193-215. · Zbl 1485.93637
[41]	ZHANG S Q, CHEN Z-Q. Stochastic maximum principle for subdiffusions and its applications [J]. SIAM J Control Optim, 2024, 62(2): 953-981. 分数阶时间导数方程和反常亚扩散过程 –纪念茆诗松教授 · Zbl 1539.93205
[42]	陈振庆 (美国华盛顿大学数学系, 西雅图, WA 98195)
[43]	摘要: 本文介绍并且改进和推广了时间导数为分数阶的方程, 以及与反常亚扩散过程相关联的最近的一些结果. 关键词: 分数阶时间导数; 时间为分数阶的方程; 隶属子; 逆隶属子; 强解和弱解中图分类号: O211.6; O211.9; O175

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.