A question about Infinitesimal generator of Feller Process

Question

Let $S=% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion $, and consider the Feller process $\left( X_{t}\right) _{t\geq 0}$ with state space $S$ such that $X_{t}=t+X_{0}$ for all $t\geq 0$. Let $A$ be the infinitesimal generator of $\left( X_{t}\right) _{t\geq 0}.$ Show that $% D\left( A\right) =\left\{ f\in C\left( S\right) :f^{\prime }\in C\left( S\right) \right\} $ and $Af=f^{\prime }$ for all $f\in D\left( A\right) .$

Answer 1

This follows directly from the definition of the infinitesimal generator, and from the definition of the derivative.

1. Derive the semigroup $P_t$ associated to $X_t$. Prove that $P_tf(x) = f(x+t)$ for any Borel $f$.

2. By the definition of the infinitesimal generator, $$ \mathscr Af(x):=\lim_{t\downarrow 0}\frac{P_tf(x) - f(x)}{t} = \lim_{t\downarrow 0}\frac{f(x+t) - f(x)}{t} = f'(x) \tag{1} $$ whenever $f$ has a derivative at $x$. As a result, you see that to belong to $D(\mathscr A)$ the function has to have a derivative pointwise everywhere.

3. Since $D(\mathscr A)$ is the set of all functions for which the convergence in $(1)$ is uniform, you get that $f'$ has to be a continuous function. Indeed, you have that $g(x,t) := \frac{f(x+t) - f(x)}{t}$ that are continuous functions converge uniformly to $f'$. As a result, the latter function has to be continuous as well.

I have an impression though, that $f'$ also has to be bounded - which you may wanna check by yourself.