We prove a Feynman-Kac-type formula for the heat flow acting on 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 Weitzenbock operator acting on forms and the second fundamental form of the boundary. As a geometric application we find a new obstruction to the existence of metrics with positive isotropic curvature and 2-convex boundary.We also present a version of the Feynman-Kac formula for spinors under suitable boundary conditions and discuss potential applications.