  This paper proposes a probabilistic approach to investigate the shape of the
landscape of multi-dimensional potential functions. By using an appropriate
coupling scheme, two copies of the overdamped Langevin dynamics of the
potential function are coupled. For unimodal and multimodal potential
functions, the tail distributions of the coupling times are shown to have
qualitatively different dependency on the noise magnitude. More specifically,
for a single-well potential function that is strongly convex, the exponential
tail of the coupling time distribution is independent of noise and uniformly
bounded away from zero by the convexity parameter; while for a multi-well
potential function, the exponential tail is exponentially small with respect to
the noise magnitude with the exponents determined by the essential barrier
height, a quantity proposed in this paper which, in a sense, characterizes the
``non-convexity" of the potential function. This provides a promising approach
to detect the shape of a potential landscape through the coupling time
distributions which, in certain sense, shares the similar spirit with the
well-know problem ``Can one hear the shape of a drum?" proposed by Kac in his
famous paper in 1966. Theoretical results are numerically verified and
discussed for a variety of examples in different contexts, including the
Rosenbrock function, the potential of interacting particle systems, as well as
the loss functions of artificial neural networks.
