# American Institute of Mathematical Sciences

February  2020, 19(2): 641-676. doi: 10.3934/cpaa.2020030

## Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates

 1 Department of Mathematics, The Chinese University of Hong Kong, China 2 Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China 3 Key Laboratory of HPCSIP, Ministry of Education of China, College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan 410081, China 4 Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA 30460-8093, USA

*Corresponding author

Received  March 2017 Revised  May 2019 Published  October 2019

Fund Project: This research was supported in part by the National Natural Science Foundation of China, grants 11871296, 11771136 and 11271122, by Tsinghua University Initiative Scientific Research Program, and by a Hong Kong RGC grant. SN is also supported in part by Construct Program of the Key Discipline in Hunan Province, the Hunan Province Hundred Talents Program, and the Center of Mathematical Sciences and Applications of Harvard University

We study the geometric properties of self-similar measures on intervals generated by iterated function systems (IFS's) that do not satisfy the open set condition (OSC) and have overlaps. The examples studied in this paper are the infinite Bernoulli convolution associated with the golden ratio, and a family of convolutions of Cantor-type measures. We make use of Strichartz second-order identities defined by auxiliary IFS's to compute measures of cells on different levels. These auxiliary IFS's do satisfy the OSC and are used to define new metrics. As an application, we obtain sub-Gaussian heat kernel estimates of the time changed Brownian motions with respect to these measures. The walk dimensions obtained under these new metrics are strictly greater than $2$ and are closely related to the spectral dimension of fractal Laplacians.

Citation: Qingsong Gu, Jiaxin Hu, Sze-Man Ngai. Geometry of self-similar measures on intervals with overlaps and applications to sub-Gaussian heat kernel estimates. Communications on Pure & Applied Analysis, 2020, 19 (2) : 641-676. doi: 10.3934/cpaa.2020030
##### References:
 [1] M. T. Barlow and R. F. Bass, Transition densities for Brownian motion on the Sierpiński carpet, Probab. Theory Related Fields, 91 (1992), 307-330.  doi: 10.1007/BF01192060.  Google Scholar [2] M. T. Barlow and R. F. Bass, Brownian motion and harmonic analysis on Sierpiński carpets, Canad. J. Math., 51 (1999), 673-744.  doi: 10.4153/CJM-1999-031-4.  Google Scholar [3] 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), 1642-1677.  doi: 10.1002/cpa.20091.  Google Scholar [4] M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-624.  doi: 10.1007/BF00318785.  Google Scholar [5] E. J. Bird, S.-M. Ngai and A. Teplyaev, Fractal Laplacians on the unit interval, Ann. Sci. Math. Québec, 27 (2003), 135-168.   Google Scholar [6] K. Dalrymple, R. S. Strichartz and J. P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.  doi: 10.1007/BF01261610.  Google Scholar [7] M. Elekes, T. Keleti and A. Máthé, Self-similar and self-affine sets: measure of the intersection of two copies, Ergodic Theory Dynam. Systems, 30 (2010), 399-440.  doi: 10.1017/S0143385709000121.  Google Scholar [8] P. J. Fitzsimmons, B. M. Hambly and T. Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys., 165 (1994), 595-620.   Google Scholar [9] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, second revised and extended edition, Walter de Gruyter, Studies in Mathematics, 19, 2011.  Google Scholar [10] A. Grigor'yan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math., 66 (2014), 641-699.  doi: 10.4153/CJM-2012-061-5.  Google Scholar [11] A. Grigor'yan, J. Hu and K.-S. Lau, Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces, J. Math. Soc. Japan, 67 (2015), 1485-1549.  doi: 10.2969/jmsj/06741485.  Google Scholar [12] A. Grigor'yan, J. Hu and K.-S. Lau, Heat kernels on metric-measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc., 355 (2003), 2065-2095.  doi: 10.1090/S0002-9947-03-03211-2.  Google Scholar [13] A. Grigor'yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab., 40 (2012), 1212-1284.  doi: 10.1214/11-AOP645.  Google Scholar [14] B.M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 79 (1999), 431–458. doi: 10.1112/S0024611599001744.  Google Scholar [15] J. Hu, An analytical approach to heat kernel estimates on strongly recurrent metric spaces, Proc. Edin. Math. Soc., 51 (2008), 171-199.  doi: 10.1017/S001309150500177X.  Google Scholar [16] J. Hu, K.-S. Lau and S.-M. Ngai, Laplace operators related to self-similar measures on $\mathbb{R}^d$, J. Funct. Anal., 239 (2006), 542-565.  doi: 10.1016/j.jfa.2006.07.005.  Google Scholar [17] J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc., 199 (2009), no. 932. doi: 10.1090/memo/0932.  Google Scholar [18] J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), no. 1015. doi: 10.1090/S0065-9266-2011-00632-5.  Google Scholar [19] J. Kigami, Time changes of the Brownian motion: Poincaré inequality, heat kernel estimate and protodistance, Mem. Amer. Math. Soc., 209 (2019), no. 1250.  Google Scholar [20] T. Kumagai, Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Related Fields, 96 (1993), 205-224.  doi: 10.1007/BF01192133.  Google Scholar [21] T. Kumagai, Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms, Publ. Res. Inst. Math. Sci., 40 (2004), 793-818.   Google Scholar [22] T. Kumagai and K. T. Sturm, Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces, J. Math. Kyoto Univ., 45 (2005), 307-327.  doi: 10.1215/kjm/1250281992.  Google Scholar [23] K.-S. Lau and S.-M. Ngai, Second-order self-similar identities and multifractal decompositions, Indiana Univ. Math. J., 49 (2000), 925-972.  doi: 10.1512/iumj.2000.49.1789.  Google Scholar [24] Y.-T. Lee, Infinite propagation speed for wave solutions on some p.c.f. fractals, preprint. Google Scholar [25] S.-M. Ngai, Spectral asymptotics of Laplacians associated with one-dimensional iterated function systems with overlaps, Canad. J. Math., 63 (2011), 648-688.  doi: 10.4153/CJM-2011-011-3.  Google Scholar [26] S.-M. Ngai, W. Tang and Y. Xie, Wave propagation speed on fractals, preprint. Google Scholar [27] R. S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc., 46 (1999), 1199-1208.   Google Scholar [28] R. S. Strichartz, A. Taylor and T. Zhang, Densities of self-similar measures on the line, Experiment. Math., 4 (1995), 101-128.   Google Scholar

show all references

##### References:
 [1] M. T. Barlow and R. F. Bass, Transition densities for Brownian motion on the Sierpiński carpet, Probab. Theory Related Fields, 91 (1992), 307-330.  doi: 10.1007/BF01192060.  Google Scholar [2] M. T. Barlow and R. F. Bass, Brownian motion and harmonic analysis on Sierpiński carpets, Canad. J. Math., 51 (1999), 673-744.  doi: 10.4153/CJM-1999-031-4.  Google Scholar [3] 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), 1642-1677.  doi: 10.1002/cpa.20091.  Google Scholar [4] M. T. Barlow and E. A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields, 79 (1988), 543-624.  doi: 10.1007/BF00318785.  Google Scholar [5] E. J. Bird, S.-M. Ngai and A. Teplyaev, Fractal Laplacians on the unit interval, Ann. Sci. Math. Québec, 27 (2003), 135-168.   Google Scholar [6] K. Dalrymple, R. S. Strichartz and J. P. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl., 5 (1999), 203-284.  doi: 10.1007/BF01261610.  Google Scholar [7] M. Elekes, T. Keleti and A. Máthé, Self-similar and self-affine sets: measure of the intersection of two copies, Ergodic Theory Dynam. Systems, 30 (2010), 399-440.  doi: 10.1017/S0143385709000121.  Google Scholar [8] P. J. Fitzsimmons, B. M. Hambly and T. Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys., 165 (1994), 595-620.   Google Scholar [9] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, second revised and extended edition, Walter de Gruyter, Studies in Mathematics, 19, 2011.  Google Scholar [10] A. Grigor'yan and J. Hu, Heat kernels and Green functions on metric measure spaces, Canad. J. Math., 66 (2014), 641-699.  doi: 10.4153/CJM-2012-061-5.  Google Scholar [11] A. Grigor'yan, J. Hu and K.-S. Lau, Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces, J. Math. Soc. Japan, 67 (2015), 1485-1549.  doi: 10.2969/jmsj/06741485.  Google Scholar [12] A. Grigor'yan, J. Hu and K.-S. Lau, Heat kernels on metric-measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc., 355 (2003), 2065-2095.  doi: 10.1090/S0002-9947-03-03211-2.  Google Scholar [13] A. Grigor'yan and A. Telcs, Two-sided estimates of heat kernels on metric measure spaces, Ann. Probab., 40 (2012), 1212-1284.  doi: 10.1214/11-AOP645.  Google Scholar [14] B.M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3), 79 (1999), 431–458. doi: 10.1112/S0024611599001744.  Google Scholar [15] J. Hu, An analytical approach to heat kernel estimates on strongly recurrent metric spaces, Proc. Edin. Math. Soc., 51 (2008), 171-199.  doi: 10.1017/S001309150500177X.  Google Scholar [16] J. Hu, K.-S. Lau and S.-M. Ngai, Laplace operators related to self-similar measures on $\mathbb{R}^d$, J. Funct. Anal., 239 (2006), 542-565.  doi: 10.1016/j.jfa.2006.07.005.  Google Scholar [17] J. Kigami, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc., 199 (2009), no. 932. doi: 10.1090/memo/0932.  Google Scholar [18] J. Kigami, Resistance forms, quasisymmetric maps and heat kernel estimates, Mem. Amer. Math. Soc., 216 (2012), no. 1015. doi: 10.1090/S0065-9266-2011-00632-5.  Google Scholar [19] J. Kigami, Time changes of the Brownian motion: Poincaré inequality, heat kernel estimate and protodistance, Mem. Amer. Math. Soc., 209 (2019), no. 1250.  Google Scholar [20] T. Kumagai, Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Related Fields, 96 (1993), 205-224.  doi: 10.1007/BF01192133.  Google Scholar [21] T. Kumagai, Heat kernel estimates and parabolic Harnack inequalities on graphs and resistance forms, Publ. Res. Inst. Math. Sci., 40 (2004), 793-818.   Google Scholar [22] T. Kumagai and K. T. Sturm, Construction of diffusion processes on fractals, $d$-sets, and general metric measure spaces, J. Math. Kyoto Univ., 45 (2005), 307-327.  doi: 10.1215/kjm/1250281992.  Google Scholar [23] K.-S. Lau and S.-M. Ngai, Second-order self-similar identities and multifractal decompositions, Indiana Univ. Math. J., 49 (2000), 925-972.  doi: 10.1512/iumj.2000.49.1789.  Google Scholar [24] Y.-T. Lee, Infinite propagation speed for wave solutions on some p.c.f. fractals, preprint. Google Scholar [25] S.-M. Ngai, Spectral asymptotics of Laplacians associated with one-dimensional iterated function systems with overlaps, Canad. J. Math., 63 (2011), 648-688.  doi: 10.4153/CJM-2011-011-3.  Google Scholar [26] S.-M. Ngai, W. Tang and Y. Xie, Wave propagation speed on fractals, preprint. Google Scholar [27] R. S. Strichartz, Analysis on fractals, Notices Amer. Math. Soc., 46 (1999), 1199-1208.   Google Scholar [28] R. S. Strichartz, A. Taylor and T. Zhang, Densities of self-similar measures on the line, Experiment. Math., 4 (1995), 101-128.   Google Scholar
(a) The IFS {S0, S1} has overlaps. (b) The auxiliary IFS {T0, T1, T2} does not have overlaps
Points $x, y$ and cells $K_{\omega ^{\prime }}, K_{\tau ^{\prime }}$, where $\omega = \omega ^{\prime }0$
Positions of three points $x, y, z$ when $\omega ^{\prime} = \omega$
A chain of $k+1$ cells with length $| \omega |$ contained in $[x, z]$
(a) The IFS $\{S_i\}_{i = 0}^3$ has overlaps. (b) The auxiliary IFS $\{T_i\}_{i = 0}^2$ does not have overlaps
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