Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 66, Number 1 (2014), 289-316.
On general boundary conditions for one-dimensional diffusions with symmetry
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].
J. Math. Soc. Japan, Volume 66, Number 1 (2014), 289-316.
First available in Project Euclid: 24 January 2014
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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