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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 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 for small times , and can be viewed as the derivative of at time 0, . As the transition function is likely not to be differentiable in any strong sense, the generator is only defined on some subset of .

Let be a Feller transition function on the lccb space E. Then, is said to be in the domain of the infinitesimal generator if the limit

exists under the uniform topology on

The operator is called the infinitesimal generator of the semigroup .

[This] can alternatively be written as

[…]So, the generator A gives the first-order approximation to for small t.

Restricted to , the operator is differentiable with derivative given by . Equation (8) is a version of the Kolmogorov backward equation.

OK, so now what can I do with this? TBC.