## On general boundary conditions for one-dimensional diffusions with symmetry.(English)Zbl 1296.60211

Let $$X^0$$ be a minimal diffusion on a one-dimensional open interval $$I$$. The author studies the $$m$$-symmetry of part processes of $$X^0$$ on each relatively compact subinterval of $$I$$ and calculates corresponding symmetric forms via integration by parts. Then, all possible symmetric diffusion extensions of $$X^0$$ in terms of boundary conditions are characterized both in the $$L^2$$-setting and the $$C_b$$-setting, where $$C_b$$ denotes the space of all bounded (finely) continuous functions.
Finally, the author studies symmetric diffusion extensions of $$X^0$$, which exhaust all possible symmetric diffusion-extensions of $$X^0$$ obtained by adding regular boundary points or their identification to $$I$$.

### MSC:

 60J60 Diffusion processes 60J50 Boundary theory for Markov processes 31C25 Dirichlet forms
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### References:

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