It is shown that, for systems with an arbitrary number of degrees of freedom, a necessary and sufficient condition for the Wigner function to be nonnegative is that the corresponding state wavefunction is the exponential of a quadratic form. This result generalizes the one obtained by Hudson [Rep. Math. Phys. 6, 249 (1974)] for one-dimensional systems.
It is shown, following a criterion borrowed from Khas’minskii, that the stochastic process associated with the (approximate) Fokker–Planck equation of the hydrogen atom problem in stochastic electrodynamics (SED) is nonrecurrent and therefore also nonergodic. The demonstration of this nonrecurrence property does not use any explicit solution. The property implies, among other things, that all the invariant measures of the process will be nonfinite. Some remarks concerning the consequences for SED are made.
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