  In this work, we consider a wide two-layer neural network and study the
behavior of its empirical weights under a dynamics set by a stochastic gradient
descent along the quadratic loss with mini-batches and noise. Our goal is to
prove a trajectorial law of large number as well as a central limit theorem for
their evolution. When the noise is scaling as 1/N $\beta$ and 1/2 < $\beta$
$\le$ $\infty$, we rigorously derive and generalize the LLN obtained for
example in [CRBVE20, MMM19, SS20b]. When 3/4 < $\beta$ $\le$ $\infty$, we also
generalize the CLT (see also [SS20a]) and further exhibit the effect of
mini-batching on the asymptotic variance which leads the fluctuations. The case
$\beta$ = 3/4 is trickier and we give an example showing the divergence with
time of the variance thus establishing the instability of the predictions of
the neural network in this case. It is illustrated by simple numerical
examples.
