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Time Changes of the Brownian Motion: Poincaré Inequality, Heat Kernel Estimate and Protodistance
About this Title
Jun Kigami, Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan
Publication: Memoirs of the American Mathematical Society
Publication Year:
2019; Volume 259, Number 1250
ISBNs: 978-1-4704-3620-9 (print); 978-1-4704-5255-1 (online)
DOI: https://doi.org/10.1090/memo/1250
Published electronically: April 18, 2019
Keywords: Sierpinski carpet,
Brownian motion,
time change,
Poincaré inequality,
protodistance,
volume doubling property,
walk dimension,
heat kernel
MSC: Primary 31E05, 60J35, 60J60; Secondary 28A80, 30L10, 43A99, 60J65, 80A20
Table of Contents
Chapters
- 1. Introduction
- 2. Generalized Sierpinski carpets
- 3. Standing assumptions and notations
- 4. Gauge function
- 5. The Brownian motion and the Green function
- 6. Time change of the Brownian motion
- 7. Scaling of the Green function
- 8. Resolvents
- 9. Poincaré inequality
- 10. Heat kernel, existence and continuity
- 11. Measures having weak exponential decay
- 12. Protodistance and diagonal lower estimateof heat kernel
- 13. Proof of Theorem 1.1
- 14. Random measures having weak exponential decay
- 15. Volume doubling measure and sub-Gaussian heat kernel estimate
- 16. Examples
- 17. Construction of metrics from gauge function
- 18. Metrics and quasimetrics
- 19. Protodistance and the volume doubling property
- 20. Upper estimate of $p_{\mu }(t, x, y)$
- 21. Lower estimate of $p_{\mu }(t, x, y)$
- 22. Non existence of super-Gaussian heat kernel behavior
Abstract
In this paper, time changes of the Brownian motions on generalized Sierpinski carpets including $n$-dimensional cube $[0, 1]^n$ are studied. Intuitively time change corresponds to alteration to density of the medium where the heat flows. In case of the Brownian motion on $[0, 1]^n$, density of the medium is homogeneous and represented by the Lebesgue measure. Our study includes densities which are singular to the homogeneous one. We establish a rich class of measures called measures having weak exponential decay. This class contains measures which are singular to the homogeneous one such as Liouville measures on $[0, 1]^2$ and self-similar measures. We are going to show the existence of time changed process and associated jointly continuous heat kernel for this class of measures. Furthermore, we obtain diagonal lower and upper estimates of the heat kernel as time tends to $0$. In particular, to express the principal part of the lower diagonal heat kernel estimate, we introduce “protodistance”associated with the density as a substitute of ordinary metric. If the density has the volume doubling property with respect to the Euclidean metric, the protodistance is shown to produce metrics under which upper off-diagonal sub-Gaussian heat kernel estimate and lower near diagonal heat kernel estimate will be shown.- Sebastian Andres and Naotaka Kajino, Continuity and estimates of the Liouville heat kernel with applications to spectral dimensions, Probab. Theory Related Fields 166 (2016), no. 3-4, 713–752. MR 3568038, DOI 10.1007/s00440-015-0670-4
- Matthias Arbeiter and Norbert Patzschke, Random self-similar multifractals, Math. Nachr. 181 (1996), 5–42. MR 1409071, DOI 10.1002/mana.3211810102
- Martin T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995) Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1–121. MR 1668115, DOI 10.1007/BFb0092537
- Martin T. Barlow and Richard F. Bass, The construction of Brownian motion on the Sierpiński carpet, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 3, 225–257 (English, with French summary). MR 1023950
- Martin T. Barlow and Richard F. Bass, Local times for Brownian motion on the Sierpiński carpet, Probab. Theory Related Fields 85 (1990), no. 1, 91–104. MR 1044302, DOI 10.1007/BF01377631
- M. T. Barlow and R. F. Bass, On the resistance of the Sierpiński carpet, Proc. Roy. Soc. London Ser. A 431 (1990), no. 1882, 345–360. MR 1080496, DOI 10.1098/rspa.1990.0135
- Martin T. Barlow and Richard F. Bass, Transition densities for Brownian motion on the Sierpiński carpet, Probab. Theory Related Fields 91 (1992), no. 3-4, 307–330. MR 1151799, DOI 10.1007/BF01192060
- Martin T. Barlow and Richard F. Bass, Coupling and Harnack inequalities for Sierpiński carpets, Bull. Amer. Math. Soc. (N.S.) 29 (1993), no. 2, 208–212. MR 1215306, DOI 10.1090/S0273-0979-1993-00424-5
- Martin T. Barlow and Richard F. Bass, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999), no. 4, 673–744. MR 1701339, DOI 10.4153/CJM-1999-031-4
- Martin T. Barlow, Richard F. Bass, Takashi Kumagai, and Alexander Teplyaev, Uniqueness of Brownian motion on Sierpiński carpets, J. Eur. Math. Soc. (JEMS) 12 (2010), no. 3, 655–701. MR 2639315, DOI 10.4171/jems/211
- Martin T. Barlow and Takashi Kumagai, Transition density asymptotics for some diffusion processes with multi-fractal structures, Electron. J. Probab. 6 (2001), no. 9, 23. MR 1831804, DOI 10.1214/EJP.v6-82
- Richard F. Bass, Probabilistic techniques in analysis, Probability and its Applications (New York), Springer-Verlag, New York, 1995. MR 1329542
- Richard F. Bass, A stability theorem for elliptic Harnack inequalities, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 3, 857–876. MR 3085094, DOI 10.4171/JEMS/379
- Richard F. Bass, Moritz Kassmann, and Takashi Kumagai, Symmetric jump processes: localization, heat kernels and convergence, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 1, 59–71 (English, with English and French summaries). MR 2641770, DOI 10.1214/08-AIHP201
- Zhen-Qing Chen and Masatoshi Fukushima, Symmetric Markov processes, time change, and boundary theory, London Mathematical Society Monographs Series, vol. 35, Princeton University Press, Princeton, NJ, 2012. MR 2849840
- M. Deza and P. Chebotarev, Protometrics, preprint.
- Michel Marie Deza and Elena Deza, Encyclopedia of distances, 3rd ed., Springer, Heidelberg, 2014. MR 3243690, DOI 10.1007/978-3-662-44342-2
- K. J. Falconer, The multifractal spectrum of statistically self-similar measures, J. Theoret. Probab. 7 (1994), no. 3, 681–702. MR 1284660, DOI 10.1007/BF02213576
- A. H. Frink, Distance functions and the metrization problem, Bull. Amer. Math. Soc. 43 (1937), no. 2, 133–142. MR 1563501, DOI 10.1090/S0002-9904-1937-06509-8
- Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
- Christophe Garban, Rémi Rhodes, and Vincent Vargas, Liouville Brownian motion, Ann. Probab. 44 (2016), no. 4, 3076–3110. MR 3531686, DOI 10.1214/15-AOP1042
- Christophe Garban, Rémi Rhodes, and Vincent Vargas, On the heat kernel and the Dirichlet form of Liouville Brownian motion, Electron. J. Probab. 19 (2014), no. 96, 25. MR 3272329, DOI 10.1214/ejp.v19-2950
- Siegfried Graf, R. Daniel Mauldin, and S. C. Williams, The exact Hausdorff dimension in random recursive constructions, Mem. Amer. Math. Soc. 71 (1988), no. 381, x+121. MR 920961, DOI 10.1090/memo/0381
- A. Grigor’yan, Heat kernel upper bounds on fractal spaces, preprint 2004.
- Alexander Grigor’yan, Heat kernels and function theory on metric measure spaces, Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) Contemp. Math., vol. 338, Amer. Math. Soc., Providence, RI, 2003, pp. 143–172. MR 2039954, DOI 10.1090/conm/338/06073
- Alexander Grigor’yan, Jiaxin Hu, and Ka-Sing Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc. 355 (2003), no. 5, 2065–2095. MR 1953538, DOI 10.1090/S0002-9947-03-03211-2
- Alexander Grigor’yan and András Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324 (2002), no. 3, 521–556. MR 1938457, DOI 10.1007/s00208-002-0351-3
- Alexander Grigor’yan and Andras Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab. 40 (2012), no. 3, 1212–1284. MR 2962091, DOI 10.1214/11-AOP645
- Juha Heinonen, Lectures on analysis on metric spaces, Universitext, Springer-Verlag, New York, 2001. MR 1800917, DOI 10.1007/978-1-4613-0131-8
- Masanori Hino, On short time asymptotic behavior of some symmetric diffusions on general state spaces, Potential Anal. 16 (2002), no. 3, 249–264. MR 1885762, DOI 10.1023/A:1014033208581
- Masanori Hino and José A. Ramírez, Small-time Gaussian behavior of symmetric diffusion semigroups, Ann. Probab. 31 (2003), no. 3, 1254–1295. MR 1988472, DOI 10.1214/aop/1055425779
- Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042, DOI 10.1017/CBO9780511470943
- Jun Kigami, Local Nash inequality and inhomogeneity of heat kernels, Proc. London Math. Soc. (3) 89 (2004), no. 2, 525–544. MR 2078700, DOI 10.1112/S0024611504014807
- Jun Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc. 199 (2009), no. 932, viii+94. MR 2512802, DOI 10.1090/memo/0932
- Jun Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc. 216 (2012), no. 1015, vi+132. MR 2919892, DOI 10.1090/S0065-9266-2011-00632-5
- Jun Kigami, Quasisymmetric modification of metrics on self-similar sets, Geometry and analysis of fractals, Springer Proc. Math. Stat., vol. 88, Springer, Heidelberg, 2014, pp. 253–282. MR 3276005, DOI 10.1007/978-3-662-43920-3_{9}
- P. Maillard, R. Rhodes, V. Vargas, and O. Zeitouni, Liouville heat kernel: regularity and bounds, Ann. Inst. Henri Poincaré Probab. Stat. 52 (2016), no. 3, 1281–1320 (English, with English and French summaries). MR 3531710, DOI 10.1214/15-AIHP676
- N. Patzschke and U. Zähle, Self-similar random measures. IV. The recursive construction model of Falconer, Graf, and Mauldin and Williams, Math. Nachr. 149 (1990), 285–302. MR 1124811, DOI 10.1002/mana.19901490122
- Viktor Schroeder, Quasi-metric and metric spaces, Conform. Geom. Dyn. 10 (2006), 355–360. MR 2268484, DOI 10.1090/S1088-4173-06-00155-X
- Takashi Kumagai and Karl-Theodor Sturm, Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces, J. Math. Kyoto Univ. 45 (2005), no. 2, 307–327. MR 2161694, DOI 10.1215/kjm/1250281992
- P. Tukia and J. Väisälä, Quasisymmetric embeddings of metric spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. 5 (1980), no. 1, 97–114. MR 595180, DOI 10.5186/aasfm.1980.0531
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften, vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913