Moment estimates, exponential integrability, concentration inequalities and exit times estimates on evolving manifolds

On a smooth (not necessarily compact) manifold M equipped with a C¹-family of complete Riemannian metrics g (t) and a C^1, -family of vector fields Z (t) both indexed by the real interval 0, T) where T (0, , we prove moment estimates, exponential integrability, concentration inequalities and exit times estimates for diffusions on complete evolving Riemannian manifolds.