As written here, we have $f \in C_0$ in general. As discussed in comments, I wonder why/how $f \in C^2_0 $ for Brownian motion? I'd appreciate some explanation for the choice of space in the context of the definition of the infinitesimal generator.
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1"Does it mean that the path of the Brownian motion is twice differentiable?" *Definitely not.* The generator of a Markov process is a functional analytic object related to its semi group. I'd appreciate some googling before asking for basic explanations. – Kurt G. Jan 05 '24 at 12:55
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Thanks. I edited that. However, the main question is why/how $f\in C^2_0$ was defined. – Denis Jan 05 '24 at 13:00
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The question of yours you linked to is almost four years old and evolved around the same topic. Why/how $C^2_0$ is defined becomes pretty obvious by looking at the usual suspect: it is the space of twice continuously partial differentiable functions vanishing at infinity. – Kurt G. Jan 05 '24 at 13:44
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2the infinitesimal generator of Brownian Motion is the Laplacian, which is only defined for twice differentiable functions. – Stratos supports the strike Jan 05 '24 at 14:18
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Thank you. Why it must vanish at infinity? – Denis Jan 08 '24 at 18:58
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It simply depends on what you are interested. For general Markov theory, there is no need to look at functions that vanish at infinity. However, if you want to use stronger properties of Feller processes, you want to look at continuous functions that vanish at infinity. – Mushu Nrek Jan 09 '24 at 16:19