Journal of the Mathematical Society of Japan

On general boundary conditions for one-dimensional diffusions with symmetry

Masatoshi FUKUSHIMA

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 24 January 2014

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1390600846

Digital Object Identifier
doi:10.2969/jmsj/06610289

Mathematical Reviews number (MathSciNet)
MR3161402

Zentralblatt MATH identifier
1296.60211

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

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

Citation

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. https://projecteuclid.org/euclid.jmsj/1390600846


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References

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