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 C1,∞-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.