L\'evy Processes in a Step 3 Nilpotent Lie Group

TL;DR

This paper extends the analysis of Le9vy processes to step 3 nilpotent Lie groups, showing their generators are pseudo-differential operators via Weyl calculus, generalizing classical Euclidean results.

Contribution

It introduces a framework for analyzing Le9vy process generators on step 3 nilpotent Lie groups using pseudo-differential operators and Weyl calculus, expanding beyond Euclidean space.

Findings

Generators are pseudo-differential operators with $C_c(G)$ as core.

Extension of pseudo-differential operator theory to step 3 nilpotent Lie groups.

Framework generalizes classical Fourier analysis methods.

Abstract

The infinitesimal generators of L\'evy processes in Euclidean space are pseudo-differential operators with symbols given by the L\'evy-Khintchine formula. This classical analysis relies heavily on Fourier analysis which in the case when the state space is a Lie group becomes much more subtle. Still the notion of pseudo-differential operators can be extended to connected, simply connected nilpotent Lie groups by employing the Weyl functional calculus. With respect to this definition, the generators of L\'evy processes in the simplest step 3 nilpotent Lie group $G$ are pseudo-differential operators which admit $C_c(G)$ as its core.