37 - Infinitesimal generators

Summary

Often a Markov process is specified in terms of its behavior at each point, and one wants to form a global picture of the process. This means one is given the infinitesimal generator, which is a linear operator that is an unbounded operator in general, and one wants to come up with the semigroup for the Markov process.

We will begin by looking further at semigroups and resolvents, and then define the infinitesimal generator of a semigroup. We will prove the Hille–Yosida theorem, which is the primary tool for constructing semigroups from infinitesimal generators. Then we will look at two important examples: elliptic operators in nondivergence form and Lévy processes.

Semigroup properties

Let S be a locally compact separable metric space. We will take ℬ to be a separable Banach space of real-valued functions on S. For the most part, we will take ℬ to be the continuous functions on S that vanish at infinity (with the supremum norm), although another common example is to let ℬ be the set of functions on S that are in L² with respect to some measure. We use ∥·∥ for the norm on ℬ.

For the duration of this chapter we will make the following assumption.

Assumption 37.1Suppose that P_t, t ≥ 0, are operators acting on ℬ such that

1. (1) the P_tare contractions: ∥P_tf∥≤∥f∥ for all t ≥ 0 and all f ∈ ℬ,

2. (2) the P_tform a semigroup: P_sP_t = P_t+sfor all s, t ≥ 0, and

3. (3) the P_tare strongly continuous: if f ∈ ℬ, then P_tf → f as t → 0.