  We study closed systems of particles that are subject to stochastic forces in
addition to the conservative forces. The stochastic equations of motion are set
up in such a way that the energy is strictly conserved at all times. To ensure
this conservation law, the evolution equation for the probability density is
derived using an appropriate interpretation of the stochastic equation of
motion that is not the It\^o nor the Stratonovic interpretation. The
trajectories in phase space are restricted to the surface of constant energy.
Despite this restriction, the entropy is shown to increase with time,
expressing irreversible behavior and relaxation to equilibrium. This main
result of the present approach contrasts with that given by the Liouville
equation, which also describes closed systems, but does not show
irreversibility.
