Zero noise limit for multidimensional SDEs driven by a pointy gradient

Abstract: 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.