Nonstandard definition for the generator of a standard Ito diffusion

Question

For a standard Brownian motion, the generator of the diffusion is $$ L = \frac12 \frac{d^2}{dx^2}. $$

Is there a nonstandard definition of this generator?

Answer 1

try F. Herzberg at http://link.springer.com/chapter/10.1007/978-3-642-33149-7_7