  In this article we prove the existence of Bernstein processes which we
associate in a natural way with a class of linear parabolic initial-and final
boundary value problems defined in bounded convex subsets of Euclidean space of
arbitrary dimension. Under certain conditions regarding their endpoint
distributions, we also prove that such processes become reversible Markov
diffusions. Furthermore we show that those diffusions satisfy two It\^o
equations for some suitably constructed Wiener processes, and from that
analysis derive Feynman-Kac representations for the solutions to the given
equations. We then illustrate some of our results by considering the heat
equation with Neumann boundary conditions both in a one-dimensional bounded
interval and in a two-dimensional disk.
