T. Kumagai, N. Shanmugalingam and R. Shimizu,
Finite dimensionality of Besov spaces and potential-theoretic decomposition of metric spaces.
Link to arXiv

X. Chen, T. Kumagai and J. Wang,
Quenched local limit theorem for random conductance models with long-range jumps.
Link to arXiv

J.-D. Deuschel, T. Kumagai and M. Slowik,
Gradient estimates of the heat kernel for random walks among time-dependent random conductances.
Link to arXiv

X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang,
Quantitative stochastic homogenization for random conductance models with stable-like jumps.
Link to arXiv

The author would appreciate any comments on manuscripts.

T. Kumagai, Random Walks on Disordered Media and their Scaling Limits.
Lecture Notes in Mathematics, Vol. 2101, École d'Été de Probabilités de Saint-Flour XL--2010.
Springer, New York, (2014). Corrections

Refereed Papers

104) S. Cao, Z.-Q. Chen and T. Kumagai,
On Kigami's conjecture of the embedding $\mathcal{W}^p(K)\subset C(K)$.
Proc. Amer. Math. Soc. 152 (2024), no. 8, 3393--3402. Link to arXiv
https://doi.org/10.1090/proc/16779

103) S. Andres, D.A. Croydon and T. Kumagai,
Heat kernel fluctuations and quantitative homogenization for the one-dimensional Bouchaud trap model.
Stoch. Proc. Their Appl. 172 (2024), Paper No. 104336, 20 pp. Link to arXiv
https://doi.org/10.1016/j.spa.2024.104336

102) S. Andres, D.A. Croydon and T. Kumagai,
Heat kernel fluctuations for stochastic processes on fractals and random media.
Appl. Numer. Harmon. Anal. Birkhauser/Springer, Cham, 2023, 265--281. Link to arXiv
https://doi.org/10.1007/978-3-031-37800-3_12

101) Z.-Q. Chen, T. Kumagai, L. Saloff-Coste, J. Wang and T. Zheng,
Limit theorems for some long range random walks on torsion free nilpotent groups.
SpringerBriefs Math. Springer, Cham, 2023, xiii+139 pp. Link to arXiv
https://doi.org/10.1007/978-3-031-43332-0

100) Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang,
Two-sided heat kernel estimates for symmetric diffusion processes with jumps: recent results.
Dirichlet forms and related topics, 63--83, Springer Proc. Math. Stat., 394, Springer, Singapore, 2022.
https://doi.org/10.1007/978-981-19-4672-1_5

99) Z.-Q. Chen, T. Kumagai and J. Wang,
Heat kernel estimates for general symmetric pure jump Dirichlet forms.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 23 (2022), no. 3, 1091--1140. Link to arXiv
https://doi.org/10.2422/2036-2145.202001_012

98) Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang,
Heat kernels for reflected diffusions with jumps on inner uniform domains.
Trans. Amer. Math. Soc. 375 (2022), no. 10, 6797--6841. Link to arXiv
https://doi.org/10.1090/tran/8678

97) Z.-Q. Chen, T. Kumagai, L. Saloff-Coste, J. Wang and T. Zheng,
Long range random walks and associated geometries on groups of polynomial growth.
Ann. Inst. Fourier (Grenoble) 72 (2022), no. 3, 1249--1304. Link to arXiv
https://doi.org/10.5802/aif.3515

96) M. Kassmann, K.-Y. Kim and T. Kumagai,
Heat kernel bounds for nonlocal operators with singular kernels.
J. Math. Pures Appl. (9) 164 (2022), 1--26. Link to arXiv
https://doi.org/10.1016/j.matpur.2022.05.017

95) V.-H. Can, D.A. Croydon and T. Kumagai,
Spectral dimension of simple random walk on a long-range percolation cluster.
Electron. J. Probab. 27 (2022), no. 56, 1--37. Go to EJP
https://doi.org/10.1214/22-EJP783

94) X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang,
Periodic homogenization of non-symmetric Lévy-type processes.
Ann. Probab. 49 (2021), no. 6, 2874--2921. Link to arXiv
https://doi.org/10.1214/21-AOP1518

93) Z.-Q. Chen, T. Kumagai and J. Wang,
Stability of heat kernel estimates and parabolic Harnack inequalities for general symmetric pure jump processes.
Analysis and partial differential equations on manifolds, fractals and graphs, 1--26, Adv. Anal. Geom., 3,
De Gruyter, Berlin, 2021. PDF File
https://doi.org/10.1515/9783110700763-001

92) T. Kumagai,
Anomalous behavior of random walks on disordered media.
In: Creative Complex Systems (K. Nishimura et al. (eds.)), Creative Economy, pp. 73--84,
Springer 2021. PDF File
https://doi.org/10.1007/978-981-16-4457-3_5

91) M.T. Barlow, D.A. Croydon and T. Kumagai,
Quenched and averaged tails of the heat kernel of the two-dimensional uniform spanning tree.
Probab. Theory Relat. Fields 181 (2021), no. 1-3, 57--111 (Kesten volume). Link to arXiv
https://doi.org/10.1007/s00440-021-01078-w

90) Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang,
Heat kernel upper bounds for symmetric Markov semigroups.
J. Funct. Anal. 281 (2021), no. 4, 109074, 40 pp. Link to arXiv
https://doi.org/10.1016/j.jfa.2021.109074

89) M. Biskup, X. Chen, T. Kumagai and J. Wang,
Quenched Invariance Principle for a class of random conductance models with long-range jumps.
Probab. Theory Relat. Fields 180 (2021), no. 3-4, 847--889. Link to arXiv
https://doi.org/10.1007/s00440-021-01059-z

88) V.-H. Can, R. van der Hofstad and T. Kumagai,
Glauber dynamics for Ising models on random regular graphs: cut-off and metastability.
ALEA, Lat. Am. J. Probab. Math. Stat. 18 (2021), 1441--1482. Link to arXiv
https://doi.org/10.30757/ALEA.v18-52

87) X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang,
Homogenization of symmetric stable-like processes in stationary ergodic media.
SIAM J. Math. Anal. 53 (2021), no. 3, 2957--3001. Link to arXiv
https://doi.org/10.1137/20M1326726

86) X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang,
Quenched invariance principle for long range random walks in balanced random environments.
Ann. Inst. H. Poincaré. Probab. Statist. 57 (2021), no. 4, 2243--2267. Link to arXiv
https://doi.org/10.1214/21-AIHP1150

85) Z.-Q. Chen, T. Kumagai and J. Wang,
Stability of heat kernel estimates for symmetric non-local Dirichlet forms.
Memoirs Amer. Math. Soc. 271 (2021), no. 1330. Link to arXiv
https://doi.org/10.1090/memo/1330

84) X. Chen, Z.-Q. Chen, T. Kumagai and J. Wang,
Homogenization of symmetric jump processes in random media.
Rev. Roumaine Math. Pures Appl. 66 (2021), no. 1, 83--105. PDF file
Link to the Article

83) X. Chen, T. Kumagai and J. Wang,
Random conductance models with stable-like jumps: Quenched invariance principle.
Ann. Appl. Probab. 31 (2021), no. 3, 1180--1231. Link to arXiv
https://doi.org/10.1214/20-AAP1616

82) Z.-Q. Chen, T. Kumagai and J. Wang,
Heat kernel estimates and parabolic Harnack inequalities for symmetric Dirichlet forms.
Adv. Math. 374 (2020), 107269. Link to arXiv
https://doi.org/10.1016/j.aim.2020.107269

81) X. Chen, T. Kumagai and J. Wang,
Random conductance models with stable-like jumps: heat kernel estimates and Harnack inequalities.
J. Funct. Anal. 279 (2020), no. 7, 108656, 51 pp. Link to arXiv
https://doi.org/10.1016/j.jfa.2020.108656

80) Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang,
Time fractional Poisson equations: Representations and estimates.
J. Funct. Anal. 278 (2020), no. 2, 108311, 48 pp. Link to arXiv
https://doi.org/10.1016/j.jfa.2019.108311

79) Z.-Q. Chen, T. Kumagai and J. Wang,
Stability of parabolic Harnack inequalitiess for symmetric non-local Dirichlet forms.
J. Eur. Math. Soc. 22 (2020), no. 11, 3747--3803. Link to arXiv
https://doi.org/10.4171/JEMS/996

78) D.A. Croydon, B.M. Hambly and T. Kumagai,
Heat kernel estimates for FIN processes associated with resistance forms.
Stoch. Proc. Their Appl. 129 (2019), no. 9, 2991--3017. PDF File

77) Z.-Q. Chen, T. Kumagai and J. Wang,
Elliptic Harnack inequalities for symmetric non-local Dirichlet forms.
J. Math. Pures Appl. 125 (2019), no. 9, 1--42. PDF File

76) G.-Y. Chen and T. Kumagai,
Products of random walks on finite groups with moderate growth.
Tohoku Math. J. 71 (2019), no. 2, 281--302. PDF File

75) T. Kumagai, Anomalous random walks and diffusions on disordered media (in Japanese).
"Sugaku", Iwanami-shoten, 70 (2018), no. 1, 81--100.

74) A. Dembo, T. Kumagai and C. Nakamura,
Cutoff for lamplighter chains on fractals.
Electron. J. Probab. 23 (2018), no. 73, 1--21. Go to EJP

73) Z.-Q. Chen, P. Kim, T. Kumagai and J. Wang,
Heat kernel estimates for time fractional equations.
Forum. Math., 30 (2018), no. 5, 1163--1192. PDF File

72) G.-Y. Chen and T. Kumagai,
Cutoffs for product chains.
Stoch. Proc. Their Appl., 128 (2018), no. 11, 3840--3879. PDF File

71) Z.-Q. Chen, T. Kumagai and J. Wang,
Mean value inequalities for jump processes.
Stochastic Partial Differential Equations and Related Fields,
In Honor of Michael Röckner, SPDERF, Bielefeld, pp. 421--437,
Springer Proc. in Math and Stat., 229 (2018). PDF File

70) T. Kumagai and C. Nakamura,
Lamplighter random walks on fractals.
J. Theoret. Probab. 31 (2018), no. 1, 68--92. PDF File

69) D.A. Croydon, B.M. Hambly and T. Kumagai,
Time-changes of stochastic processes associated with resistance forms.
Electron. J. Probab. 22 (2017), no. 82, 1--41. Go to EJP

68) P. Kim, T. Kumagai and J. Wang,
Laws of the iterated logarithm for symmetric jump processes.
Bernoulli 23 (2017), no. 4A, 2330-2379. PDF File

67) M.T. Barlow, D.A. Croydon and T. Kumagai,
Subsequential scaling limits of simple random walk on the two-dimensional uniform spanning tree.
Ann. Probab. 45 (2017), no. 1, 4--55. (Revised Version) PDF File

66) T. Kumagai and C. Nakamura,
Laws of the iterated logarithm for random walks on Random Conductance Models.
RIMS Kôkyûroku Bessatsu B59 (2016), 141--156. PDF File

65) R. Huang and T. Kumagai,
Stability and instability of Gaussian heat kernel estimates for random walks among time-dependent conductances.
Electron. Commun. Probab. 2016, Vol. 21, paper no. 5, 1-11. Go to ECP

64) O. Boukhadra, T. Kumagai and P. Mathieu,
Harnack inequalities and local central limit theorem for the polynomial lower tail random conductance model.
J. Math. Soc. Japan, 67 (2015), no. 4, 1413--1448. (Revised Version) PDF File

63) Z.-Q. Chen, D.A. Croydon and T. Kumagai,
Quenched invariance principles for random walks and elliptic diffusions in random media with boundary.
Ann. Probab. 43 (2015), no. 4, 1594--1642. (Revised Version) PDF File

62) K. Bogdan, T. Kumagai and M. Kwaśnicki,
Boundary Harnack inequality for Markov processes with jumps.
Trans. Amer. Math. Soc. 367 (2015), no. 1 , 477--517. (Revised Version) PDF File

61) T. Kumagai, Anomalous random walks and diffusions: From fractals to random media.
Proceedings of the ICM Seoul 2014, Vol. IV, 75--94, Kyung Moon SA Co. Ltd. 2014. PDF File

60) T. Kumagai and Zeitouni,
Fluctuations of recentered maxima of discrete Gaussian Free Fields on a class of recurrent graphs.
Electron. Commun. Probab., 18 (2013), no. 75, 1--12. Go to ECP

59) D.A. Croydon, A. Fribergh and T. Kumagai,
Biased random walk on critical Galton-Watson trees conditioned to survive.
Probab. Theory Relat. Fields, 157 (2013), 453--507. (Revised Version) PDF File (277kb)

58) J.-D. Deuschel and T. Kumagai,
Markov chain approximations to non-symmetric diffusions with bounded coefficients.
Comm. Pure Appl. Math. 66 (2013), no. 6, 821--866. (Revised Version) PDF File (292kb)

57) Z.-Q. Chen, P. Kim and T. Kumagai,
Discrete Approximation of Symmetric Jump Processes on Metric Measure Spaces.
Probab. Theory Relat. Fields 155 (2013), 703--749. (Revised Version) PDF File (430kb)

56) M.T. Barlow, A. Grigor'yan and T. Kumagai,
On the equivalence of parabolic Harnack inequalities and heat kernel estimates.
J. Math. Soc. Japan, 64 (2012), no. 4, 1091--1146. (Revised Version) PDF File (714kb)

55) D.A. Croydon, B.M. Hambly and T. Kumagai,
Convergence of mixing times for sequences of random walks on finite graphs.
Electron. J. Probab., 17 (2012), no. 3, 1--32. Go to EJP

54) Z.-Q. Chen, P. Kim and T. Kumagai,
Global Heat Kernel Estimates for Symmetric Jump Processes.
Trans. Amer. Math. Soc., 363 (2011), no. 9, 5021--5055. (Revised Version) PDF File (569kb) PS File (1897kb)
Corrections

53) R.F. Bass, T. Kumagai and T. Uemura,
Convergence of symmetric Markov chains on Z^d.
Probab. Theory Relat. Fields, 148 (2010), 107--140. (Revised Version) PDF File (252kb) PS File (552kb) Correction PDF File (68Kb)

52) Z.-Q. Chen and T. Kumagai,
A priori Hölder estimate, parabolic Harnack principle and heat kernel estimates for diffusions with jumps.
Rev. Mat. Iberoamericana, 26 (2010), 551--589. PDF File (289kb)

51) M.T. Barlow, R.F. Bass, T. Kumagai and A. Teplyaev,
Uniqueness of Brownian motion on Sierpinski carpets.
J. European Math. Soc., 12 (2010), 655--701. PDF File (368kb) PS File (761kb)

* Supplementary notes for "Uniqueness of Brownian motion on Sierpinski carpets"
(joint with M.T. Barlow, R.F. Bass and A. Teplyaev), PDF File (223kb) PS File (518kb)

50) B.M. Hambly and T. Kumagai,
Diffusion on the scaling limit of the critical percolation cluster in the diamond hierarchical lattice.
Comm. Math. Phys., 295 (2010), 29--69. PDF File (392kb) PS File (1966kb)

49) R.F. Bass, M. Kassmann and T. Kumagai,
Symmetric jump processes: localization, heat kernels, and convergence.
Ann. Inst. H. Poincaré - Probabilités et Statistiques, 46 (2010), 59--71. PDF File (160kb) PS File (365kb)

48) Z.-Q. Chen, P. Kim and T. Kumagai,
On Heat kernel estimates and parabolic Harnack inequality for jump processes on metric measure spaces.
Acta Math. Sin. (Engl. Ser.) 25 (2009), 1067--1086. PDF File (206kb) PS File (485kb)

47) M.T. Barlow, R.F. Bass and T. Kumagai,
Parabolic Harnack inequality and heat kernel estimates for random walks with long range jumps.
Math. Z. 261 (2009), no. 2, 297--320. PDF File (208kb), Post Script File. (458kb) Corrections PDF File

46) M.T. Barlow, A. Grigor'yan and T. Kumagai,
Heat kernel upper bounds for jump processes and the first exit time.
J. Reine Angew. Math. 626 (2009), 135--157. PDF File (318kb), Post Script File. (779kb) Corrections PDF File (104Kb)

45) T. Kumagai and J. Misumi,
Heat kernel estimates for strongly recurrent random walk on random media.
J. Theoret. Probab. 21 (2008), no. 4, 910--935. (Revised Version) PDF File (221kb), Post Script File. (498kb)

44) Z.-Q. Chen, P. Kim and T. Kumagai,
Weighted Poincaré inequality and heat kernel estimates for finite range jump processes.
Math. Ann., 342, (2008), no. 4, 833--883. PDF File (278kb), PS File (1366kb)

43) A. Grigor'yan anf T. Kumagai,
On the dichotomy in the heat kernel two sided estimates.
In: Analysis on Graphs and its Applications (P. Exner et al. (eds.)), Proc. of Symposia in Pure Math. 77,
pp. 199--210, Amer. Math. Soc. 2008. PDF File (169kb), PS File (454kb)

42) D. Croydon and T. Kumagai,
Random walks on Galton-Watson trees with infinite variance offspring distribution conditioned to survive.
Electron. J. Probab., 13 (2008), 1419--1441. Go to EJP

41) T. Kumagai,
Recent developments of analysis on fractals.
Translations, Series 2, Volume 223, pp. 81--95, Amer. Math. Soc. 2008.
PDF File (381kb), Post Script File. (2270kb)

40) M.T. Barlow, A.A. Járai, T. Kumagai and G. Slade,
Random walk on the incipient infinite cluster for oriented percolation in high dimensions.
Comm. Math. Phys., 278 (2008), no 2, 385--431.
PDF File (434kb), Post Script File. (897kb)

39) I. Fujii and T. Kumagai,
Heat kernel estimates on the incipient infinite cluster for critical branching processes.
Proceedings of German-Japanese symposium in Kyoto 2006,
RIMS Kôkyûroku Bessatsu B6 (2008), 85--95 PDF File (138kb), Post Script File. (359kb)

38) R.F. Bass and T. Kumagai,
Symmetric Markov chains on Z^d with unbounded range.
Trans. Amer. Math. Soc., 360 (2008), no. 4, 2041--2075. PDF File (525kb), Post Script File. (558kb)

37) Z.-Q. Chen and T. Kumagai,
Heat kernel estimates for jump processes of mixed types on metric measure spaces.
Probab. Theory Relat. Fields, 140 (2008), no. 1-2, 277--317. PDF File (440kb), Post Script File. (517kb)

36) J. Hu and T. Kumagai,
Nash-type inequalities and heat kernels for non-local Dirichlet forms.
Kyushu J. Math., 60 (2006), no.2, 245--265. Post Script File. (328kb)

35) M.T. Barlow and T. Kumagai,
Random walk on the incipient infinite cluster on trees.
Illinois J. Math., 50 (2006), no.1, 33--65. (Doob volume) PDF File (308kb),Post Script File. (414kb)

34) M. Hino and T. Kumagai,
A trace theorem for Dirichlet forms on fractals.
J. Funct. Anal., 238 (2006), no.2, 578--611. PDF File (613kb),Post Script File. (2021kb)
Corrections.

33) M.T. Barlow, R.F. Bass and T. Kumagai,
Stability of parabolic Harnack inequalities on metric measure spaces.
J. Math. Soc. Japan, 58 (2006), no. 2, 485--519. PDF File (401kb), Post Script File. (1431kb)
Corrections.

*Note on the equivalence of parabolic Harnack inequalities and heat kernel estimates
(joint with M.T. Barlow and R.F. Bass), Post Script File. (235kb)

32) M.T. Barlow, T. Coulhon and T. Kumagai,
Characterization of sub-Gaussian heat kernel estimates on strongly recurrent graphs.
Comm. Pure Appl. Math., 58 (2005), no. 12, 1642--1677. PDF File (249kb), Post Script File. (357kb)

31) K.T. Sturm and T. Kumagai,
Construction of diffusion processes on fractals, d-sets, and general metric measure spaces.
J. Math. Kyoto Univ. 45 (2005), no. 2, 307--327. Post Script File. (317kb)

30) B.M. Hambly and T. Kumagai,
Heat kernel estimates for symmetric random walks on a class of fractal graphs and stability under rough isometries.
In: Fractal geometry and applications: A Jubilee of B. Mandelbrot (M.L. Lapidus and M. van Frankenhuijsen (eds.)),
Proc. of Symposia in Pure Math. 72, Part 2, pp. 233--260, Amer. Math. Soc. 2004. PDF File (343kb), Post Script File. (540kb)

29) T. Kumagai,
Recent developments of analysis on fractals (in Japanese).
"Sugaku", Iwanami-shoten, 56 (2004), no.4, 337--350.

28) T. Kumagai,
Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms.
Publ. RIMS, Kyoto Univ., 40 (2004), 793--818. Post Script File. (309kb) Corrections PDF File. (757Kb)

27) T. Kumagai,
Function spaces and stochastic processes on fractals.
In: Fractal geometry and stochastics III (C. Bandt et al. (eds.)), Progr. Probab. 57, pp. 221--234,
Birkhauser, 2004. Post Script File. (267kb)

26) B.M. Hambly and T. Kumagai,
Heat kernel estimates and law of the iterated logarithm for symmetric random walks on fractal graphs.
In: Discrete Geometric Analysis, (M. Kotani et al. (eds.)), Contemporary Mathematics 347, pp. 153--172,
Amer. Math. Soc. 2004. PDF File (301kb), Post Script File. (351kb)

25) T. Kumagai,
Homogenization on finitely ramified fractals.
Advanced Studies in Pure Math., 41, Stochastic Analysis and Related Topics in Kyoto (H. Kunita et al. (eds.)),
pp. 189--207, MSJ, 2004. PDF File (248kb), Post Script File. (270kb)

24) B.M. Hambly and T. Kumagai,
Diffusion processes on fractal fields: heat kernel estimates and large deviations.
Probab. Theory Relat. Fields, 127 (2003), no.3, 305--352.
PDF File (609kb), Post Script File. (1936kb)

23) Z.-Q. Chen and T. Kumagai,
Heat kernel estimates for stable-like processes on d-sets.
Stoch. Proc. Their Appl., 108 (2003), no. 1, 27--62. PDF File (363kb), Post Script File. (459kb)

22) T. Kumagai,
Some remarks for stable-like jump processes on fractals.
In: Trends in Math., Fractals in Graz 2001 (P. Grabner and W. Woess (eds.)),
pp. 185-196, Birkhauser, 2002. Post Script File (210kb)

21) B.M. Hambly and T. Kumagai,
Asymptotics for the spectral and walk dimension as fractals approach Euclidean space.
Fractals, 10 (2002), no. 4, 403--412. PDF File. (230kb)

20) R.F. Bass and T. Kumagai,
Laws of the iterated logarithm for the range of random walks in two and three dimensions.
Ann. Probab., 30 (2002), no. 3, 1369--1396. Reprint

19) B.M. Hambly, J. Kigami and T. Kumagai,
Multifractal formalisms for the local spectral and walk dimensions.
Math. Proc. Cambridge Philos. Soc., 132 (2002), no. 3, 555--571. PDF file. (Revised Draft)

18) M.T. Barlow and T. Kumagai,
Transition density asymptotics for some diffusion processes with multi-fractal structures.
Electronic Journal of Probability, (paper 9) 6 (2001), 1--23. Go to EJP

17) B.M. Hambly and T. Kumagai,
Fluctuation of the transition density for Brownian motion on random recursive Sierpinski gaskets.
Stoch. Proc. Their Appl., 92 (2001), no. 1, 61--85. Post Script File. (439kb)

16) R.F. Bass and T. Kumagai,
Laws of the iterated logarithm for some symmetric diffusion processes.
Osaka J. Math., 37 (2000), no. 3, 625--650. Reprint

15) B.M. Hambly, T. Kumagai, S. Kusuoka and X.Y. Zhou,
Transition density estimates for diffusion processes on homogeneous random Sierpinski carpets.
J. Math. Soc. Japan, 52 (2000), no. 2, 373--408. Post Script File. (1643Kb)

14) T. Kumagai,
Stochastic processes on fractals and related topics.
Sugaku Expositions, Amer. Math. Soc., 13 (2000), no. 1, 55--71.

13) G. Ben Arous and T. Kumagai,
Large deviations for Brownian motion on the Sierpinski gasket.
Stoch. Proc. Their Appl., 85 (2000), 225--235. Reprint

12) T. Kumagai,
Brownian motion penetrating fractals -An application of the trace theorem of Besov spaces-.
J. Funct. Anal., 170 (2000), no. 1, 69--92. Reprint Corrections PDF File.

11) B.M. Hambly and T. Kumagai,
Transition density estimates for diffusion processes on p.c.f. self-similar fractals.
Proc. London Math. Soc., 78 (1999), no. 3, 431--458.

10) B.M. Hambly and T. Kumagai,
Heat kernel estimates and homogenization for asymptotically lower dimensional processes on some nested fractals.
Potential Anal., 8 (1998), 359--397.

9) T. Kumagai,
Stochastic processes on fractals and related topics (in Japanese).
"Sugaku", Iwanami-shoten, 49 (1997), no. 2, 158--172.

8) T. Kumagai,
Short time asymptotic behavior and large deviations for Brownian motion on some affine nested fractals.
Publ. RIMS. Kyoto Univ., 33 (1997), 223--240.

7) T. Kumagai,
Percolation on pre-Sierpinski carpets.
In: New trends in stochastic analysis -Proceedings of a Taniguchi International Workshop (K.D.Elworthy et al (eds.)),
World Scientific, 1997, pp. 288-304.

6) T. Kumagai and S. Kusuoka,
Homogenization on nested fractals.
Probab. Theory Relat. Fields, 104 (1996), 375--398.

5) T. Kumagai,
Rotation invariance and characterization of a class of self-similar diffusion processes on the Sierpinski gasket.
In: Algorithms, fractals, and dynamics (Y.Takahashi (ed.)), Plenum, 1995, pp. 131--142.

4) P.J. Fitzsimmons, B.M. Hambly and T. Kumagai,
Transition density estimates for Brownian motion on affine nested fractals.
Comm. Math. Phys., 165 (1994), no. 3, 595--620.

3) T. Kumagai,
Estimates of transition densities for Brownian motion on nested fractals. (Ph.D. thesis)
Probab. Theory Relat. Fields, 96 (1993), 205--224.

2) T. Kumagai,
Regularity, closedness and spectral dimensions of the Dirichlet forms on P.C.F. self-similar sets.
J. Math. Kyoto Univ., 33 (1993), 765--786.

1) T. Kumagai,
Construction and some properties of a class of non-symmetric diffusion processes on the Sierpinski gasket.
In: Asymptotic problems in probability theory: stochastic models and diffusions on fractals (Elworthy, K.D. and Ikeda, N.
(eds.)), Pitman, 1993, pp. 219--247.