Over the path space of a symplectic manifold with end points in two La
grangian submanifolds, we define a measure and a stochastic symplectic
action in the simply connected case. We define a regularized Wess-Zum
ino-Witten Laplacian over the forms of finite degree over the path spa
ce. We perform a short time asymptotic near the critical points and fi
nd a limit Brownian harmonic oscillator: we can diagonalize it explici
tly, and find the limit ground state of the Laplacian. We define a sto
chastic Witten complex, and its algebraic counterpart at the level of
Chen forms.