The Annals of Probability

Brownian motion on some spaces with varying dimension

Zhen-Qing Chen and Shuwen Lou

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this paper, we introduce and study Brownian motion on a class of state spaces with varying dimension. Starting with a concrete case of such state spaces that models a big square with a flag pole, we construct a Brownian motion on it and study how heat propagates on such a space. We derive sharp two-sided global estimates on its transition density function (also called heat kernel). These two-sided estimates are of Gaussian type, but the measure on the underlying state space does not satisfy volume doubling property. Parabolic Harnack inequality fails for such a process. Nevertheless, we show Hölder regularity holds for its parabolic functions. We also derive the Green function estimates for this process on bounded smooth domains. Brownian motion on some other state spaces with varying dimension are also constructed and studied in this paper.

Article information

Source
Ann. Probab., Volume 47, Number 1 (2019), 213-269.

Dates
Received: May 2016
Revised: October 2017
First available in Project Euclid: 13 December 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1544691621

Digital Object Identifier
doi:10.1214/18-AOP1260

Mathematical Reviews number (MathSciNet)
MR3909969

Zentralblatt MATH identifier
07036337

Subjects
Primary: 60J60: Diffusion processes [See also 58J65] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]
Secondary: 31C25: Dirichlet spaces 60H30: Applications of stochastic analysis (to PDE, etc.) 60J45: Probabilistic potential theory [See also 31Cxx, 31D05]

Keywords
Space of varying dimension Brownian motion Laplacian transition density function heat kernel estimates Hölder regularity Green function

Citation

Chen, Zhen-Qing; Lou, Shuwen. Brownian motion on some spaces with varying dimension. Ann. Probab. 47 (2019), no. 1, 213--269. doi:10.1214/18-AOP1260. https://projecteuclid.org/euclid.aop/1544691621


Export citation

References

  • [1] Andres, S. and Barlow, M. T. (2015). Energy inequalities for cutoff functions and some applications. J. Reine Angew. Math. 699 183–215.
    Zentralblatt MATH: 1347.31003
  • [2] Aronson, D. G. (1968). Non-negative solutions of linear parabolic equations. Ann. Sc. Norm. Super. Pisa (3) 22 607–694.
    Zentralblatt MATH: 0182.13802
  • [3] Barlow, M. T. and Bass, R. F. (2004). Stability of parabolic Harnack inequalities. Trans. Amer. Math. Soc. 356 1501–1533.
    Zentralblatt MATH: 1034.60070
    Digital Object Identifier: doi:10.1090/S0002-9947-03-03414-7
  • [4] Barlow, M. T., Bass, R. F. and Kumagai, T. (2006). Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Japan 58 485–519.
    Zentralblatt MATH: 1102.60064
    Digital Object Identifier: doi:10.2969/jmsj/1149166785
    Project Euclid: euclid.jmsj/1149166785
  • [5] Bass, R. F. and Chen, Z.-Q. (2005). One-dimensional stochastic differential equations with singular and degenerate coefficients. Sankhyā 67 19–45.
    Zentralblatt MATH: 1192.60081
  • [6] Bass, R. F. and Chen, Z.-Q. (2010). Regularity of harmonic functions for a class of singular stable-like processes. Math. Z. 266 489–503.
    Mathematical Reviews (MathSciNet): MR2719417
    Zentralblatt MATH: 1201.60055
    Digital Object Identifier: doi:10.1007/s00209-009-0581-0
  • [7] Biroli, M. and Mosco, U. (1995). A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. (4) 169 125–181.
    Mathematical Reviews (MathSciNet): MR1378473
    Zentralblatt MATH: 0851.31008
    Digital Object Identifier: doi:10.1007/BF01759352
  • [8] Carlen, E. A., Kusuoka, S. and Stroock, D. W. (1987). Upper bounds for symmetric Markov transition functions. Ann. Inst. Henri Poincaré Probab. Stat. 23 245–287.
    Zentralblatt MATH: 0634.60066
  • [9] Chen, Z.-Q. Topics on recent developments in the theory of Markov processes. Available at http://www.math.washington.edu/~zchen/RIMS_lecture.pdf.
  • [10] Chen, Z.-Q. and Fukushima, M. (2011). Symmetric Markov Processes, Time Change, and Boundary Theory. London Mathematical Society Monographs Series 35. Princeton Univ. Press, Princeton, NJ.
    Mathematical Reviews (MathSciNet): MR2849840
  • [11] Chen, Z.-Q., Fukushima, M. and Rohde, S. (2016). Chordal Komatu–Loewner equation and Brownian motion with darning in multiply connected domains. Trans. Amer. Math. Soc. 368 4065–4114.
    Zentralblatt MATH: 1374.60123
    Digital Object Identifier: doi:10.1090/tran/6441
  • [12] Cho, S., Kim, P. and Park, H. (2012). Two-sided estimates on Dirichlet heat kernels for time-dependent parabolic operators with singular drifts in $C^{1,\alpha}$-domains. J. Differential Equations 252 1101–1145.
    Zentralblatt MATH: 1233.35107
    Digital Object Identifier: doi:10.1016/j.jde.2011.07.025
  • [13] Chung, K. L. and Zhao, Z. (1995). Form Brownian Motion to Schrödinger’s Equation. Springer, Berlin.
  • [14] Delmotte, T. (1999). Parabolic Harnack inequality and estimates of Markov chains on graphs. Rev. Mat. Iberoam. 15 181–232.
    Mathematical Reviews (MathSciNet): MR1681641
    Zentralblatt MATH: 0922.60060
    Digital Object Identifier: doi:10.4171/RMI/254
  • [15] Evans, S. N. and Sowers, R. B. (2003). Pinching and twisting Markov processes. Ann. Probab. 31 486–527.
    Mathematical Reviews (MathSciNet): MR1959800
    Zentralblatt MATH: 1017.60084
    Digital Object Identifier: doi:10.1214/aop/1046294318
    Project Euclid: euclid.aop/1046294318
  • [16] Fukushima, M., Ōshima, Y. and Takeda, M. (2011). Dirichlet Forms and Symmetric Markov Processes, 2nd rev. and ext. ed. de Gruyter, Berlin.
    Zentralblatt MATH: 1227.31001
  • [17] Grigor’yan, A. (1994). Heat kernel upper bounds on a complete non-compact manifold. Rev. Mat. Iberoam. 10 395–452.
  • [18] Grigor’yan, A. and Saloff-Coste, L. (2002). Hitting probabilities for Brownian motion on Riemannian manifolds. J. Math. Pures Appl. (9) 81 115–142.
    Zentralblatt MATH: 1042.58022
    Digital Object Identifier: doi:10.1016/S0021-7824(01)01244-2
  • [19] Hambly, B. M. and Kumagai, T. (2003). Diffusion processes on fractal fields: Heat kernel estimates and large deviations. Probab. Theory Related Fields 127 305–352.
    Zentralblatt MATH: 1035.60083
    Digital Object Identifier: doi:10.1007/s00440-003-0284-0
  • [20] Hansen, W. (2017). Darning and gluing of diffusions. Potential Anal. 46 167–180.
    Zentralblatt MATH: 1362.60067
    Digital Object Identifier: doi:10.1007/s11118-016-9577-7
  • [21] Hsu, E. P. (2002). Stochastic Analysis on Manifolds. Graduate Studies in Mathematics 38. Amer. Math. Soc., Providence, RI.
    Zentralblatt MATH: 0994.58019
  • [22] Karatzas, I. and Shreve, S. E. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics 113. Springer, New York.
    Zentralblatt MATH: 0638.60065
  • [23] Kumagai, T. (2000). Brownian motion penetrating fractals: An application of the trace theorem of Besov spaces. J. Funct. Anal. 170 69–92.
    Zentralblatt MATH: 0960.31005
    Digital Object Identifier: doi:10.1006/jfan.1999.3500
  • [24] Liu, J. S. and Sabatti, C. (1999). Simulated sintering: Markov chain Monte Carlo with spaces of varying dimensions. In Bayesian Statistics, 6 (Alcoceber, 1998) 389–413. Oxford Univ. Press, New York.
    Zentralblatt MATH: 0974.62026
  • [25] Mörters, P. and Peres, Y. (2010). Brownian Motion. Cambridge Series in Statistical and Probabilistic Mathematics 30. Cambridge Univ. Press, Cambridge.
  • [26] Revuz, D. and Yor, M. (1991). Continuous Martingales and Brownian Motion. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin.
  • [27] Saloff-Coste, L. (1992). A note on Poincaré, Sobolev, and Harnack inequalities. Int. Math. Res. Not. 2 27–38.
    Mathematical Reviews (MathSciNet): MR1150597
    Zentralblatt MATH: 0769.58054
    Digital Object Identifier: doi:10.1155/S1073792892000047
  • [28] Sturm, K.-T. (1995). Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 275–312.
    Zentralblatt MATH: 0854.35015
    Project Euclid: euclid.ojm/1200786053
  • [29] Sturm, K. T. (1996). Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. (9) 75 273–297.
    Zentralblatt MATH: 0854.35016
  • [30] Uchiyama, K. (2012). Asymptotic estimates of the distribution of Brownian hitting time of a disc. J. Theoret. Probab. 25 450–463. Erratum: J. Theoret. Probab. 25 (2012) 910–911.
    Zentralblatt MATH: 1255.60147
    Digital Object Identifier: doi:10.1007/s10959-010-0305-8
  • [31] Xia, C. (2007). Optimal control of switched systems with dimension-varying state spaces. Ph.D dissertation in electrical engineering, Univ. California, Los Angeles.
  • [32] Zhang, Q. S. (1997). Gaussian bounds for the fundamental solutions of $\nabla(A\nabla u)+B\nabla u-u_{t}=0$. Manuscripta Math. 93 381–390.
    Zentralblatt MATH: 0886.35004
    Digital Object Identifier: doi:10.1007/BF02677479
  • [33] Zhang, Q. S. (2003). The global behavior of heat kernels in exterior domains. J. Funct. Anal. 200 160–176.
    Zentralblatt MATH: 1021.58019
    Digital Object Identifier: doi:10.1016/S0022-1236(02)00074-5

The Institute of Mathematical Statistics

The Annals of Probability

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more