# Feller processes, infinitesimal generators

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 generator is 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 $$\{{P_t}\}_{t \geq 0}$$ be a Feller transition function on the lccb space E. Then, $${f\in C_0(E)}$$ is said to be in the domain $${\mathcal{D}_A}$$ of the infinitesimal generator if the limit

$Af=\lim_{t\rightarrow0}\frac1t(P_tf-f)$

exists under the uniform topology on $$\{C_0(E)}.$$

The operator $${A\colon\mathcal{D}_A\rightarrow C_0(E)}$$ is called the infinitesimal generator of the semigroup $${\{P_t\}_{t\ge 0}}$$.

[This] can alternatively be written as

$P_tf = f + tAf + \mathcal{o}(t)$

[…]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? TBC.