This page exists because no one could explain to me why I should care about infinitesimal generators. Then I found George Lowther:

[Feller Processes] are Markov processes whose transition function \(\{P_t\}_{t\ge 0}\) satisfies certain continuity conditions.[…] […]it is often not possible to explicitly write out the transition function describing a Feller process. Instead, the

infinitesimal generatoris used. This approximately describes the transition kernel \(P_t\) for small times \(t\), and can be viewed as the derivative of \(P_t\) at time 0, \(A=dP_t/dt\vert_{t=0}\). As the transition function is likely not to be differentiable in any strong sense, the generator is only defined on some subset of \(C_0\).Let {{P_t}_{tge0}} be a Feller transition function on the lccb space E. Then, {fin C_0(E)} is said to be in the domain {mathcal{D}_A} of the infinitesimal generator if the limit

\begin{equation*} Af=\lim_{t\rightarrow0}\frac1t(P_tf-f) \end{equation*}exists under the uniform topology on {C_0(E)}.

The operator {Acolonmathcal{D}_Arightarrow C_0(E)} is called the infinitesimal generator of the semigroup {{P_t}_{tge 0}}.

[This] can alternatively be written as

\begin{equation*} P_tf = f + tAf + o(t) \end{equation*}where {o(t)} denotes a term vanishing faster than t as {trightarrow0}. So, the generator A gives the first-order approximation to {P_t} for small t.

Restricted to {mathcal{D}_A}, the operator {P_t} is differentiable with derivative given by {AP_t=P_tA}. Equation (8) is a version of the Kolmogorov backward equation.

OK, so now what can I do with this?