Quantum diffusion on manifolds

We investigate non-degenerate diffusion processes on an arbitrary manifold, the dynamics of which arise from a principle of least action for a Lagrangian consisting of a kinetic term quadratic in the forward drift of the process and a local potential. The equation governing the action emerges as a stochastic Hamilton–Jacobi condition and is expressed in terms of the geometry determined by the Levi-Civita connection of the diffusion tensor. It is argued that there are essentially two dynamical structures for the rate of change of the drift in the presence of a local potential, consistent with the requirement of time reversal symmetry. In both cases a conserved energy is identified. An alternative wave function and associated operator description reveals a complex structure in the dynamical equations, thus extending the earlier results of Nelson on the stochastic treatment of the Schrödinger equation.