Accelerating diffusions

Let U be a given function defined on ℝd and π(x) be a density function proportional to exp−U(x). The following diffusion X(t) is often used to sample from π(x), dX(t)=-∇ U(X(t))\,dt+√2\,dW(t), X(0)=x₀. To accelerate the convergence, a family of diffusions with π(x) as their common equilibrium is considered, dX(t)=(-∇ U(X(t))+C(X(t)))\,dt+√2\,dW(t), X(0)=x₀. Let LC be the corresponding infinitesimal generator. The spectral gap of LC in L2(π) (λ(C)), and the convergence exponent of X(t) to π in variational norm (ρ(C)), are used to describe the convergence rate, where λ(C)=Sup {real part of μ:μ is in the spectrum of LC, μ is not zero}, ρ(C)=Inf\ρ: ∫ p(t,x,y)-π(y) \,dy≤ g(x)eρ t\. Roughly speaking, LC is a perturbation of the self-adjoint L0 by an antisymmetric operator C⋅∇, where C is weighted divergence free. We prove that λ(C)≤λ(0) and equality holds only in some rare situations. Furthermore, ρ(C)≤λ(C) and equality holds for C=0. In other words, adding an extra drift, C(x), accelerates convergence. Related problems are also discussed.