  Probability distributions which can be obtained from superpositions of
Gaussian distributions of different variances v = \sigma ^2 play a favored role
in quantum theory and financial markets. Such superpositions need not
necessarily obey the Chapman-Kolmogorov semigroup relation for Markovian
processes because they may introduce memory effects. We derive the general form
of the smearing distributions in v which do not destroy the semigroup property.
The smearing technique has two immediate applications. It permits simplifying
the system of Kramers-Moyal equations for smeared and unsmeared conditional
probabilities, and can be conveniently implemented in the path integral
calculus. In many cases, the superposition of path integrals can be evaluated
much easier than the initial path integral. Three simple examples are
presented, and it is shown how the technique is extended to quantum mechanics.
