Lévy-Type Processes and Pseudodifferential Operators

Our aim in this survey is to show how pseudodifferential operators arise naturally in the theory of Markov processes and that this opens the way to use Fourier analytic techniques for general Markov processes. In order to keep things simple, we will only consider ℝ n as state space. We will see in Section 1 that one can identify the characteristic exponent ψ(ξ) of a Lévy process with the symbol of its infinitesimal generator which is a pseudodifferential operator with constant coefficients. The key observation now is that we can—under reasonable assumptions (e.g., Feller)— associate with any Markov process {X t } t≥0 on ℝ n a function q(x, ξ) which turns out to be the symbol of the generator for {X t } t≥0 which is a pseudodifferential operator. This function q(x, ξ) is given by $$ q(x,\xi ): = - \mathop {\lim }\limits_{t \to 0} {{{E^x}({e^{i({X_t} - x) \cdot \xi }}) - 1} \over t}. $$ Needless to say that this formula will produce ψ(ξ) in the case of a Lévy process. Our programme is simple to state: study the process through its symbol.