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Heat kernel estimates for time fractional equations

  • Zhen-Qing Chen , Panki Kim , Takashi Kumagai and Jian Wang EMAIL logo
Published/Copyright: February 16, 2018

Abstract

In this paper, we establish existence and uniqueness of weak solutions to general time fractional equations and give their probabilistic representations. We then derive sharp two-sided estimates for fundamental solutions of a family of time fractional equations in metric measure spaces.


Communicated by Christopher D. Sogge


Award Identifier / Grant number: 2016R1E1A1A01941893

Award Identifier / Grant number: 11522106

Award Identifier / Grant number: 26

Award Identifier / Grant number: 2015J01003

Funding statement: The research of Panki Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1E1A1A01941893). The research of Takashi Kumagai is supported by the Grant-in-Aid for Scientific Research (A) 25247007 and 17H01093, Japan. The research of Jian Wang is supported by National Natural Science Foundation of China (No. 11522106), the JSPS postdoctoral fellowship (2604021), Fok Ying Tung Education Foundation (No. 151002), National Science Foundation of Fujian Province (No. 2015J01003), the Program for Probability and Statistics: Theory and Application (No. IRTL1704), and Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications (FJKLMAA).

References

[1] B. Baeumer and M. M. Meerschaert, Stochastic solutions for fractional Cauchy problems, Fract. Calc. Appl. Anal. 4 (2001), no. 4, 481–500. Search in Google Scholar

[2] M. T. Barlow and J. Černý, Convergence to fractional kinetics for random walks associated with unbounded conductances, Probab. Theory Related Fields 149 (2011), no. 3–4, 639–673. 10.1007/s00440-009-0257-zSearch in Google Scholar

[3] D. Brockmann, L. Hufnagel and T. Geisel, The scaling laws of human travel, Nature 439 (2006), 462–465. 10.1038/nature04292Search in Google Scholar PubMed

[4] J. Černý, On two-dimensional random walk among heavy-tailed conductances, Electron. J. Probab. 16 (2011), no. 10, 293–313. 10.1214/EJP.v16-849Search in Google Scholar

[5] Z.-Q. Chen, Time fractional equations and probabilistic representation, Chaos Solitons Fractals 102 (2017), 168–174. 10.1016/j.chaos.2017.04.029Search in Google Scholar

[6] Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, London Math. Soc. Monogr. Ser. 35, Princeton University Press, Princeton, 2012. 10.23943/princeton/9780691136059.001.0001Search in Google Scholar

[7] Z.-Q. Chen and T. Kumagai, Heat kernel estimates for jump processes of mixed types on metric measure spaces, Probab. Theory Related Fields 140 (2008), no. 1–2, 277–317. 10.1007/s00440-007-0070-5Search in Google Scholar

[8] Z.-Q. Chen, T. Kumagai and J. Wang, Stability of heat kernel estimates for symmetric jump processes on metric measure spaces, preprint (2016), https://arxiv.org/abs/1604.04035. Search in Google Scholar

[9] S. D. Eidelman and A. N. Kochubei, Cauchy problem for fractional diffusion equations, J. Differential Equations 199 (2004), no. 2, 211–255. 10.1016/j.jde.2003.12.002Search in Google Scholar

[10] M. Foondun and E. Nane, Asymptotic properties of some space-time fractional stochastic equations, Math. Z. 287 (2017), no. 1–2, 493–519. 10.1007/s00209-016-1834-3Search in Google Scholar

[11] A. Grigor’yan and T. Kumagai, On the dichotomy in the heat kernel two sided estimates, Analysis on Graphs and Its Applications, Proc. Sympos. Pure Math. 77, American Mathematical Society, Providence (2008), 199–210. 10.1090/pspum/077/2459870Search in Google Scholar

[12] B. M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. Lond. Math. Soc. (3) 78 (1999), no. 2, 431–458. 10.1112/S0024611599001744Search in Google Scholar

[13] N. C. Jain and W. E. Pruitt, Lower tail probability estimates for subordinators and nondecreasing random walks, Ann. Probab. 15 (1987), no. 1, 75–101. 10.1214/aop/1176992257Search in Google Scholar

[14] P. Kim and A. Mimica, Green function estimates for subordinate Brownian motions: Stable and beyond, Trans. Amer. Math. Soc. 366 (2014), no. 8, 4383–4422. 10.1090/S0002-9947-2014-06017-0Search in Google Scholar

[15] M. M. Meerschaert and H.-P. Scheffler, Limit theorems for continuous-time random walks with infinite mean waiting times, J. Appl. Probab. 41 (2004), no. 3, 623–638. 10.1239/jap/1091543414Search in Google Scholar

[16] M. M. Meerschaert and H.-P. Scheffler, Stochastic model for ultraslow diffusion, Stochastic Process. Appl. 116 (2006), no. 9, 1215–1235. 10.1016/j.spa.2006.01.006Search in Google Scholar

[17] M. M. Meerschaert and A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter Stud. Math. 43, Walter de Gruyter, Berlin, 2012. 10.1515/9783110258165Search in Google Scholar

[18] A. Mimica, Heat kernel estimates for subordinate Brownian motions, Proc. Lond. Math. Soc. (3) 113 (2016), no. 5, 627–648. 10.1112/plms/pdw043Search in Google Scholar

[19] J. Nakagawa, Personal communications. Search in Google Scholar

[20] J. Nakagawa, K. Sakamoto and M. Yamamoto, Overview to mathematical analysis for fractional diffusion equations—new mathematical aspects motivated by industrial collaboration, J. Math. Ind. 2A (2010), 99–108. Search in Google Scholar

[21] A. I. Saichev and G. M. Zaslavsky, Fractional kinetic equations: Solutions and applications, Chaos 7 (1997), no. 4, 753–764. 10.1063/1.166272Search in Google Scholar PubMed

[22] R. L. Schilling, R. Song and Z. Vondraček, Bernstein Functions. Theory and Applications, 2nd ed., De Gruyter Stud. Math. 37, Walter de Gruyter & Co., Berlin, 2010. 10.1515/9783110215311Search in Google Scholar

[23] M. F. Shlesinger, J. Klafter and Y. M. Wong, Random walks with infinite spatial and temporal moments, J. Stat. Phys. 27 (1982), no. 3, 499–512. 10.1007/BF01011089Search in Google Scholar

[24] A. Telcs, The Art of Random Walks, Lecture Notes in Math. 1885, Springer, Berlin, 2006. 10.1007/b134090Search in Google Scholar

[25] G. M. Zaslavsky, Fractional kinetic equation for Hamiltonian chaos, Phys. D 76 (1994), no. 1–3, 110–122. 10.1016/0167-2789(94)90254-2Search in Google Scholar

Received: 2017-09-09
Published Online: 2018-02-16
Published in Print: 2018-09-01

© 2018 Walter de Gruyter GmbH, Berlin/Boston

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