  Let $\Omega\subset\mathbb{R}^n$ be a strictly convex domain with smooth
boundary and diameter $D$. The fundamental gap conjecture claims that if
$V:\bar\Omega\to\mathbb{R}$ is convex, then the spectral gap of the
Schr\"odinger operator $-\Delta+V$ with Dirichlet boundary condition is greater
than $\frac{3\pi^2}{D^2}$. Using analytic methods, Andrews and Clutterbuck
recently proved in [J. Amer. Math. Soc. 24 (2011), no. 3, 899--916] a more
general spectral gap comparison theorem which implies this conjecture. In the
first part of the current work, we shall give an independent probabilistic
proof of their result via the coupling by reflection of the diffusion
processes. Moreover, we also present in the second part a simpler probabilistic
proof of the original conjecture.
