  In this article, we study the persistence of properties of a given classical
deter-ministic dierential equation under a stochastic perturbation of two
distinct forms: external and internal. The rst case corresponds to add a noise
term to a given equation using the framework of It\^o or Stratonovich
stochastic dierential equations. The second case corresponds to consider a
parameters dependent dierential equations and to add a stochastic dynamics on
the parameters using the framework of random ordinary dierential equations. Our
main concerns for the preservation of properties is stability/instability of
equilibrium points and symplectic/Poisson Hamiltonian structures. We formulate
persistence theorem in these two cases and prove that the cases of external and
internal stochastic perturbations are drastically dierent. We then apply our
results to develop a stochastic version of the Landau-Lifshitz equation. We
discuss in particular previous results obtain by Etore and al. in [P. 'Etore,
S.Labbe, J. Lelong, Long time behaviour of a stochastic nanoparticle, J.
Differential Equations 257 (2014), 2115-2135] and we nally propose a new family
of stochastic Landau-Lifshitz equations.
