  We prove that smooth solutions of non-ideal (viscous and resistive)
incompressible magnetohydrodynamic equations satisfy a stochastic law of flux
conservation. This property involves an ensemble of surfaces obtained from a
given, fixed surface by advecting it backward in time under the plasma velocity
perturbed with a random white-noise. It is shown that the magnetic flux through
the fixed surface is equal to the average of the magnetic fluxes through the
ensemble of surfaces at earlier times. This result is an analogue of the
well-known Alfven theorem of ideal MHD and is valid for any value of the
magnetic Prandtl number. A second stochastic conservation law is shown to hold
at unit Prandtl number, a random version of the generalized Kelvin theorem
derived by Bekenstein-Oron for ideal MHD. These stochastic conservation laws
are not only shown to be consequences of the non-ideal MHD equations, but are
proved in fact to be equivalent to those equations. We derive similar results
for two more refined hydromagnetic models, Hall magnetohydrodynamics and the
two-fluid plasma model, still assuming incompressible velocities and isotropic
transport coefficients. Finally, we use these results to discuss briefly the
infinite-Reynolds-number limit of hydromagnetic turbulence and to support the
conjecture that flux-conservation remains stochastic in that limit.
