Journal of the Mathematical Society of Japan

On general boundary conditions for one-dimensional diffusions with symmetry


Full-text: Open access


We give a simple proof of the symmetry of a minimal diffusion $X^0$ on a one-dimensional open interval $I$ with respect to the attached canonical measure $m$ along with the identification of the Dirichlet form of $X^0$ on $L^2(I; m)$ in terms of the triplet $(s,m,k)$ attached to $X^0$. The $L^2$-generators of $X^0$ and its reflecting extension $X^r$ are then readily described. We next use the associated reproducing kernels in connecting the $L^2$-setting to the traditional $C_b$-setting and thereby deduce characterizations of the domains of $C_b$-generators of $X^0$ and $X^r$ by means of boundary conditions. We finally identify the $C_b$-generators for all other possible symmetric diffusion extensions of $X^0$ and construct by that means all diffusion extensions of $X^0$ in [IM2].

Article information

J. Math. Soc. Japan, Volume 66, Number 1 (2014), 289-316.

First available in Project Euclid: 24 January 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60J60: Diffusion processes [See also 58J65]
Secondary: 60J50: Boundary theory 31C25: Dirichlet spaces

minimal diffusion canonical measure symmetric extension diffusion extension $C_b$-generator boundary condition


FUKUSHIMA, Masatoshi. On general boundary conditions for one-dimensional diffusions with symmetry. J. Math. Soc. Japan 66 (2014), no. 1, 289--316. doi:10.2969/jmsj/06610289.

Export citation


  • Z.-Q. Chen and M. Fukushima, Symmetric Markov Processes, Time Change, and Boundary Theory, London Math. Soc. Monogr. Ser, 35, Princeton University Press, 2012.
  • W. Feller, The parabolic differential equations and the associated semi-groups of transformations, Ann. of Math. (2), 55 (1952), 468–519.
  • W. Feller, Generalized second order differential operators and their lateral conditions, Illinois J. Math., 1 (1957), 459–504.
  • M. Fukushima, On boundary conditions for multi-dimensional Brownian motions with symmetric resolvent densities, J. Math. Soc. Japan, 21 (1969), 58–93.
  • M. Fukushima, From one dimensional diffusions to symmetric Markov processes, Stochastic Process. Appl., 120 (2010), 590–604. (Special issue A tribute to Kiyosi Itô)
  • M. Fukushima, Y. Oshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, de Gruyter Stud. Math., 19, Walter de Gruyter, 1994, 2011.
  • K. Itô, Essentials of Stochastic Processes, Second extended ed., Transl. Math. Monogr., 231, Amer. Math. Soc., Providence, RI, 2006 (originally published in Japanese, Iwanami Shoten, 1957).
  • K. Itô, Poisson point processes and their application to Markov processes, Lecture Notes of Mathematics Department, Kyoto University, September 1969 (unpublished).
  • K. Itô and H. P. McKean, Jr., Brownian motions on a half line, Illinois J. Math., 7 (1963), 181–231.
  • K. Itô and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer-Verlag, 1965; In: Classics in Mathematics, Springer-Verlag, 1996.
  • J. Ying and M. Zhao, The uniqueness of symmetrizing measure of Markov processes, Proc. Amer. Math. Soc., 138 (2010), 2181–2185.