### Book chapter

## The Brownian motion on Aff(R) and quasi-local theorems

This paper is concerned with Random walk approximations of the

Brownian motion on the Affine group Aff(R). We are in particular interested

in the case where the innovations are discrete. In this framework, the return

probabilities of the walk have fractional exponential decay in large time, as

opposed to the polynomial one of the continuous object. We prove that in

tegrating those return probabilities on a suitable neighborhood of the origin,

the expected polynomial decay is restored. This is what we call a Quasi-local

theorem.