  The purpose of the article is to address the limiting behavior of the
solutions of stochastic differential equations driven by a pointy
$d$-dimensional gradient as the intensity of the underlying Brownian motion
tends to $0$. By pointy gradient, we here mean that the drift derives from a
potential that is ${\mathcal C}^{1,1}$ on any compact subset that does not
contain the origin. As a matter of fact, the corresponding deterministic
version of the differential equation may have an infinite number of solutions
when initialized from $0_{{\mathbb R}^d}$, in which case the limit theorem
proved in the paper reads as a selection theorem of the solutions to the zero
noise system.
  Generally speaking, our result says that, under suitable conditions, the
probability that the particle leaves the origin by going through regions of
higher potential tends to $1$ as the intensity of the noise tends to $0$. In
particular, our result extends the earlier one due to Bafico and Baldi for the
zero noise limit of one dimensional stochastic differential equations.
