  We are concerned with averaging theorems for $\epsilon$-small stochastic
perturbations of integrable equations in $\mathbb{R}^d \times \mathbb{T}^n
=\{(I,\varphi)\}$ $$ \dot I(t) =0,\quad \dot \varphi(t) = \theta(I),
  \qquad (1)$$ and in $\mathbb{R}^{2n} = \{v=(\mathbf{v}_1, \dots,
\mathbf{v}_n), \; \mathbf{v}_j \in \mathbb{R}^2\}$, $$ \dot{\mathbf{v}}_k(t)
=W_k(I) \mathbf{v}_k^\bot, \quad k=1, \dots, n,
  \qquad (2) $$ where $I=(I_1, \dots, I_n)$ is the vector of actions, $I_j =
\frac12 \| \mathbf{v}_j\|^2$. The vector-functions $\theta$ and $W$ are locally
Lipschitz and non-degenerate. Perturbations of these equations are assumed to
be locally Lipschitz and such that some few first moments of the norms of their
solutions are bounded uniformly in $\epsilon$, for $0\le t\le \epsilon^{-1} T$.
For $I$-components of solutions for perturbations of (1) we establish their
convergence in law to solutions of the corresponding averaged $I$-equations,
when $0\le \tau := \epsilon t\le T$ and $\epsilon\to0$. Then we show that if
the system of averaged $I$-equations is mixing, then the convergence is uniform
in the slow time $\tau=\epsilon t\ge0$. Next using these results, for
$\epsilon$-perturbed equations of (2) we construct well posed {\it effective
stochastic equations} for $v(\tau)\in \mathbb{R}^{2n}$ (independent from
$\epsilon$) such that when $\epsilon\to0$, actions of solutions of the
perturbed equations of (2) with $t:= \tau/\epsilon$ converge in distribution to
actions of solutions for the effective equations. Again, if the effective
system is mixing, this convergence is uniform in the slow time $\tau \ge0$. We
provide easy sufficient conditions on the perturbed equations which ensure that
our results apply to their solutions.
