We derive a stochastic representation formula for solutions of heat-type equations on vector bundles with time-dependent Riemannian metric over manifolds whose Riemannian metric is time-dependent as well. As a corollary we obtain a vanishing theorem for bounded ancient solutions under a curvature condition. Our results apply in particular to the case of differential forms.
"Heat equation in vector bundles with time-dependent metric." J. Math. Soc. Japan 67 (4) 1759 - 1769, October, 2015. https://doi.org/10.2969/jmsj/06741759