We study infinite order differential operators acting in the spaces of exponential
type entire functions. We derive conditions under which such operators preserve the set of
Laguerre entire functions which consists of the polynomials possessing real nonpositive
zeros only and of their uniform limits on compact subsets of the complex plane. We obtain
integral representations of some particular cases of these operators and apply these
results to obtain explicit solutions to some Cauchy problems for diffusion equations with
nonconstant drift term.