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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.