  It is shown that superpositions of path integrals with arbitrary Hamiltonians
and different scaling parameters v ("variances") obey the Chapman-Kolmogorov
relation for Markovian processes if and only if the corresponding smearing
distributions for v have a specific functional form. Ensuing "smearing"
distributions substantially simplify the coupled system of Fokker-Planck
equations for smeared and un-smeared conditional probabilities. Simple
application in financial models with stochastic volatility is presented.
