Nelson's assignment of a Markov process in configuration space to each Schr\"odinger Hamiltonian and each pure state is generalized to mixed states, arbitrary configuration spaces given by a maximal set of commuting observables, and general Hamiltonians given by integral kernels. Conditions for the quantum-mechanical description to be reconstructable from this stochastic description are discussed. It is shown that those stochastic expectation values which are stable with respect to the operation of coupling a dynamically independent but statistically correlated system to the given one are precisely those already expressible as quantum-mechanical expectation values.