Infinitesimal generator (stochastic processes)


In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process is a partial differential operator that encodes a great deal of information about the process. The generator is used in evolution equations such as the Kolmogorov backward equation ; its L2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation.

Definition

General case

For a d-dimensional Feller process we define the generator by
whenever this limit exists in, i.e. in the space of continuous functions vanishing at infinity.
This definition parallels the one of infinitesimal generator of -semigroup.

Stochastic differential equations driven by Brownian motion

Let defined on a probability space be an Itô diffusion satisfying a stochastic differential equation of the form:
where is an m-dimensional Brownian motion and and are the drift and diffusion fields respectively. For a point, let denote the law of given initial datum, and let denote expectation with respect to.
The infinitesimal generator of is the operator, which is defined to act on suitable functions by:
The set of all functions for which this limit exists at a point is denoted, while ' denotes the set of all for which the limit exists for all. One can show that any compactly-supported function lies in ' and that:
Or, in terms of the gradient and scalar and Frobenius inner products:

Generators of some common processes