  In this paper, we investigate a continuous time version of the Stochastic
Langevin Monte Carlo method, introduced in [WT11], that incorporates a
stochastic sampling step inside the traditional over-damped Langevin diffusion.
This method is popular in machine learning for sampling posterior distribution.
We will pay specific attention in our work to the computational cost in terms
of $n$ (the number of observations that produces the posterior distribution),
and $d$ (the dimension of the ambient space where the parameter of interest is
living). We derive our analysis in the weakly convex framework, which is
parameterized with the help of the Kurdyka-\L ojasiewicz (KL) inequality, that
permits to handle a vanishing curvature settings, which is far less restrictive
when compared to the simple strongly convex case. We establish that the final
horizon of simulation to obtain an $\varepsilon$ approximation (in terms of
entropy) is of the order $( d \log(n)^2 )^{(1+r)^2} [\log^2(\varepsilon^{-1}) +
n^2 d^{2(1+r)} \log^{4(1+r)}(n) ]$ with a Poissonian subsampling of parameter
$\left(n ( d \log^2(n))^{1+r}\right)^{-1}$, where the parameter $r$ is involved
in the KL inequality and varies between $0$ (strongly convex case) and $1$
(limiting Laplace situation).
