  Recent result of the numerical simulation of stochastic motion of
conservative mechanical or weakly damped Brownian motion subject to
conservative forces reveals that, in the case of Gaussian random forces, the
path probability depends exponentially on Lagrangian action. This distribution
implies a fundamental principle generalizing the least action principle of the
Hamiltonian/Lagrangian mechanics and yields an extended formalism of mechanics
for random dynamics. Within this theory, Liouville theorem of conservation of
phase distribution breaks down. This opens a way to the Boltzmann H theorem. We
argue that the randomness is a crucial distinction between two kingdoms of
Hamiltonian/Lagrangian mechanics: the stochastic dynamics and the regular one
which is a special case of the first one for vanishing randomness. This
distinction was missing in the criticisms of this theorem from Loschmidt,
Poincar\'e and Zermelo.
