Summary: Let \(\pi(x)\) be a given probability density proportional to \(\exp(-U(x))\) in a high-dimensional Euclidean space \(\mathbb{R}^ m\). The diffusion \(dX(t)=-\nabla U(X(t))dt+\sqrt 2 dW(t)\) is often used to sample from \(\pi\). Instead of \(-\nabla U(x)\), we consider diffusions with smooth drift \(b(x)\) and having equilibrium \(\pi(x)\). First we study some general properties and then concentrate on the Gaussian case, namely, \(-\nabla U(x)=Dx\) with a strictly negative-definite real matrix \(D\) and \(b(x)=Bx\) with a stability matrix \(B\); that is, the real parts of the eigenvalues of \(B\) are strictly negative. Using the rate of convergence of the covariance of \(X(t)\) [or together with \(EX(t)]\) as the criterion, we prove that, among all such \(b(x)\), the drift \(Dx\) is the worst choice and that improvement can be made if and only if the eigenvalues of \(D\) are not identical. In fact, the convergence rate of the covariance is \(\exp(2\lambda_ M(B)t)\), where \(\lambda_ M(B)\) is the maximum of the real parts of the eigenvalues of \(B\) and the infimum of \(\lambda_ M(B)\) over all such \(B\) is \(1/m \text{tr} D\). If, for example, a “circulant” drift \(\left({\partial U\over \partial x_ m}-{\partial U\over\partial x_ 2},{\partial U\over\partial x_ 1}-{\partial U\over\partial x_ 3},\ldots,{\partial U\over\partial x_{m-1}}-{\partial U\over\partial x_ 1}\right)\) is added to \(Dx\), then for essentially all \(D\), the diffusion with the modified drift has a better convergence rate.