  In this paper we study the Cauchy problem for new classes of parabolic type
pseudodifferential equations over the rings of finite adeles and adeles. We
show that the adelic topology is metrizable and give an explicit metric. We
find explicit representations of the fundamental solutions (the heat kernels).
These fundamental solutions are transition functions of Markov processes which
are adelic analogues of the Archimedean Brownian motion. We show that the
Cauchy problems for these equations are well-posed and find explicit
representations of the evolution semigroup and formulas for the solutions of
homogeneous and non-homogeneous equations.
