  The paper study a possibility to recover a parabolic diffusion from its
time-average when the values at the initial time are unknown. This problem can
be reformulated as a new boundary value problem where a Cauchy condition is
replaced by a prescribed time-average of the solution. It is shown that this
new problem is well-posed in certain classes of solutions. The paper
establishes existence, uniqueness, and a regularity of the solution for this
new problem and its modifications, including problems with singled out terminal
values.
