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Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 28, Issue 5

Issues

Regular subspaces of skew product diffusions

Liping Li / Jiangang Ying
Published Online: 2015-10-07 | DOI: https://doi.org/10.1515/forum-2015-0012

Abstract

Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of the original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space E1×E2 and expressed as

Xt=(Xt1,XAt2),t0,

where Xi is a symmetric diffusion on Ei for i=1,2, and A is a positive continuous additive functional of X1. One of our main results indicates that any skew product type regular subspace of X, say

Yt=(Yt1,YA~t2),t0,

can be characterized as follows: the associated smooth measure of A~ is equal to that of A, and Yi corresponds to a regular subspace of Xi for i=1,2. Furthermore, we shall make some discussions on rotationally invariant diffusions on d{}, which are special skew product diffusions on (0,)×Sd-1. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on d{} to a new regular Dirichlet form on d. More precisely, fix a regular Dirichlet form (,) on the state space d. Its part Dirichlet form on d{} is denoted by (0,)0. Let (~0,~)0 be a regular subspace of (0,)0. We want to find a regular subspace (~,~) of (,) such that the part Dirichlet form of (~,~) on d{} is exactly (~0,~)0. If (~,~) exists, we call it a regular extension of (~0,~)0. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of (0,)0 has a unique regular extension.

Keywords: Regular subspaces; Dirichlet forms; skew product; rotation invariance

MSC 2010: 31C25; 60J55; 60J60

References

  • [1]

    Albeverio S., Hoegh-Krohn R. and Streit L., Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys. 18 (1977), no. 5, 907–917. Google Scholar

  • [2]

    Chen Z.-Q. and Fukushima M., One-point extensions of Markov processes by darning, Probab. Theory Related Fields 141 (2008), 61–112. Google Scholar

  • [3]

    Chen Z.-Q. and Fukushima M., Symmetric Markov Processes, Time Change, and Boundary Theory, London Math. Soc. Monogr. Ser. 35, Princeton University Press, Princeton, 2012. Google Scholar

  • [4]

    Chen Z.-Q., Fukushima M. and Ying J., Extending Markov processes in weak duality by Poisson point processes of excursions, Stochastic Analysis and Applications – The Abel Symposium 2005, Abel Symp. 2, Springer, Berlin (2007), 153–196. Google Scholar

  • [5]

    Fang X., Fukushima M. and Ying J., On regular Dirichlet subspaces of H1(I) and associated linear diffusions, Osaka J. Math. 42 (2005), no. 1, 27–41. Google Scholar

  • [6]

    Fang X., He P. and Ying J., Dirichlet Forms associated with linear diffusions, Chin. Ann. Math. Ser. B 31 (2010), no. 4, 507–518. Google Scholar

  • [7]

    Fitzsimmons P. J. and Li L., On Dirichlet forms of a solvable model in quantum mechanics, in preparation. Google Scholar

  • [8]

    Fitzsimmons P. J. and Li L., On Fukushima’s decompositions of symmetric diffusions, in preparation. Google Scholar

  • [9]

    Fukushima M., Energy forms and diffusion processes, Mathematics + Physics, Volume 1, World Scientific, Singapore (1985), 65–97. Google Scholar

  • [10]

    Fukushima M., From one dimensional diffusions to symmetric Markov processes, Stochastic Process. Appl. 120 (2010), no. 5, 590–604. Google Scholar

  • [11]

    Fukushima M. and Oshima Y., On the skew product of symmetric diffusion processes, Forum Math. 1 (1989), no. 2, 103–142. Google Scholar

  • [12]

    Fukushima M., Oshima Y. and Takeda M., Dirichlet Forms and Symmetric Markov Processes, 2nd ed., De Gruyter Stud. Math. 19, De Gruyter, Berlin, 2011. Google Scholar

  • [13]

    Fukushima M. and Tanaka H., Poisson point processes attached to symmetric diffusions, Ann. Inst. Henry Poincaré Probab. Stat. 41 (2005), no. 3, 419–459. Google Scholar

  • [14]

    Galmarino A. R., Representation of an isotropic diffusion as a skew product, Z. Wahrscheinlichkeitstheor. Verw. Geb. 1 (1963), 359–378. Google Scholar

  • [15]

    Itô K. and McKean H., Diffusion Processes and their Sample Paths, Classics Math. 125, Springer, Berlin, 1974. Google Scholar

  • [16]

    Kufner A., Weighted Sobolev Spaces, Wiley, New York, 1985. Google Scholar

  • [17]

    Kufner A. and Opic B., How to define reasonably weighted Sobolev spaces, Comment. Math. Univ. Carolin. 25 (1984), no. 3, 537–554. Google Scholar

  • [18]

    Li L. and Ying J., On structure of regular subspaces of one-dimensional Brownian motion, preprint 2014, http://arxiv.org/abs/1412.1896.

  • [19]

    Li L. and Ying J., Regular subspaces of Dirichlet forms, Festschrift Masatoshi Fukushima. In Honor of Masatoshi Fukushima’s Sanju, Interdiscip. Math. Sci. 17, World Scientific, Singapore (2015), 397–420. Google Scholar

  • [20]

    Ôkura H., Recurrence criteria for skew products of symmetric Markov processes, Forum Math. 1 (1989), no. 4, 331–357. Google Scholar

  • [21]

    Ôkura H., A new approach to the skew product of symmetric Markov processes, Mem. Fac. Engrg. Des., Kyoto Inst. Tech., Ser. Sci. Tech. 46 (1997), 1–12. Google Scholar

  • [22]

    Röckner M. and Zhang T. S., Uniqueness of generalized Schröedinger operators and applications, J. Funct. Anal. 105 (1992), no. 1, 187–231. Google Scholar

  • [23]

    Röckner M. and Zhang T. S., Uniqueness of generalized Schröedinger operators. II, J. Funct. Anal. 119 (1994), no. 2, 455–467. Google Scholar

  • [24]

    Rogers L. C. G. and Williams D., Diffusions, Markov Processes, and Martingales: Volume 2, Itô Calculus, Cambridge Math. Lib., Cambridge University Press, Cambridge, 2000. Google Scholar

  • [25]

    Turesson B. O., Nonlinear Potential Theory and Weighted Sobolev Spaces, Lecture Notes in Math. 1736, Springer, Berlin, 2000. Google Scholar

  • [26]

    Ventcel’ A. D., On lateral conditions for multidimensional diffusion processes, Teor. Veroyatnost. i Primenen. 4 (1959), 172–185. Google Scholar

  • [27]

    Zhikov V. V., On weighted Sobolev spaces, Mat. Sb. 189 (1998), no. 8, 27–58; translation in Sb. Math. 189 (1998), no. 7–8, 1139–1170. Google Scholar

About the article


Received: 2015-01-18

Revised: 2015-06-29

Published Online: 2015-10-07

Published in Print: 2016-09-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11271240

Research supported in part by NSFC grant 11271240.


Citation Information: Forum Mathematicum, Volume 28, Issue 5, Pages 857–872, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0012.

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