Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces

In this article, we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions firstly on sub-Riemannian limits of Riemannian foliations and secondly in the nonsmooth setting of $\operatorname{RCD}^{*}(K,N)$ spaces. In each case, the necessary ingredients are Ito’s formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schrodinger operators.