  We prove a Feynman-Kac formula for differential forms satisfying absolute
boundary conditions on Riemannian manifolds with boundary and of bounded
geometry. We use this to construct $L^2$ harmonic forms out of bounded ones on
the universal cover of a compact Riemannian manifold whose geometry displays a
positivity property expressed in terms of a certain stochastic average of the
Weitzenb\"ock operator $R_p$ acting on $p$-forms and the second fundamental
form of the boundary. This extends previous work by Elworthy-Li-Rosenberg on
closed manifolds to this setting. As an application we find a geometric
obstruction to the existence of metrics with 2-convex boundary and positive
$R_2$ in this stochastic sense. We also discuss a version of the Feynman-Kac
formula for spinors under suitable boundary conditions.
