The Symbol of a Markov Semimartingale

For Lévy processes it is a well known fact that there is a one-to-one correspondence between the elements of this class of processes and the so called continuous negative definite functions in the sense of Schoenberg ψ : R d −→ C. The connection between these concepts is given by
E x e i(Xt−x)′ ξ = e −tψ(ξ) .

In particular it is known that to every continuous negative definite function ψ there exists a corresponding Lévy process (Xt)t≥0. Several properties of the process can be expressed in terms of analytic properties of its characteristic exponent ψ. Within the class of (universal) Markov processes, Lévy processes are those which are stochastically continuous and homogeneous in time and space. From the perspective of stochastic modeling the last point is a rather strong restriction since it means that the process ‘behaves the same’ on every point in space and time. Therefore, it is an interesting question if there exists a function, which is somehow similar to the characteristic exponent of a Lévy process, for a larger class of Markov processes. A class to start with is the one of (nice) Feller processes, i.e. Feller processes with the property that the test functions C∞ c (Rd) are contained in the domain of their generator. In the investigation of these processes, a family of continuous negative definite functions ξ ↦−→ p(x, ξ), (x ∈ Rd ) shows up in the Fourier representation of the generator ∫ Au(x) = − e ix′ ξ ∞ p(x, ξ)û(ξ) dξ for u ∈ Cc (R d ).