Levy Processes In a Step 3 Nilpotent Lie Group

The infinitesimal generators of Lévy processes in Euclidean space are pseudo-differential operators with symbols given by the Lévy-Khintchine formula. In the absence of a canonical definition of Fourier transform which is sensible for arbitrary Lie groups, a similar characterization of these processes for Lie groups is a subtle matter. We introduce the notion of pseudo-differential operator in a connected, simply connected nilpotent Lie group G using the Weyl functional calculus. We prove that with respect to this definition, the quantized generators of Lévy processes in G are pseudo-differential operators which admit C∞cR as a core.