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If I uderstand things correctly: when we consider the generator of a Feller process $A$, we have a domain $\mathcal{D}(A) \subseteq C_0$. So, the domain of the generator is always a subset of the continuous functions with compact support, however, I have seen several places where the generator is applied to function without compact support, more specifically $f(x) = x$, $f(x) = x^2$, $f(x) = x^3$, etc. For example, applying Dynkin's formula to show that $W_t$, $W_t^2 - t$, $W_t^3 - 3 \int_0^t W_s ds$ are martingales where $W_t$ is brownian motion. How exactly is this justified?

One thing I see mentioned quite often, is the generalized generator, which is defined as the function $g$ such that $f(X_t) - f(X_0) - \int_0^tg(X_s)ds$ is a martingale. And then we define the extended infinitesimal generator as $Af := g$. However, it is not clear as to why we can conclude that $f(x) = x^n$ are such generalized generators.

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